arXiv:1001.0023v7 [math.AG] 1 Nov 2016Algebraic Geometry over C∞-rings Dominic Joyce Abstract IfXis a manifold then the R-algebra C∞(X) of smooth functions c:X→Ris aC∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φ f:C∞(X)n→C∞(X) acting by Φ f: (c1,...,c n)/mapsto→f(c1,...,c n), and these operations Φ fsatisfy many natural identities. Thus, C∞(X) actually has a far richer structure than the obviousR-algebra structure. We explain the foundations of a version of algebraic geometr y in which rings or algebras are replaced by C∞-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C∞-schemes, a category of geometric objects which generalize manifolds, and whose mor- phisms generalize smooth maps. We also study quasicoherent sheaves on C∞-schemes, and C∞-stacks, in particular Deligne–Mumford C∞-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C∞-rings and C∞-schemes have long been part of synthetic differential geometry. But we develop them in new directions. In [36–38], the author uses these tools to define d-manifolds andd-orbifolds , ‘derived’ versions of manifolds and orbifolds related to Spivak’s ‘derived manifolds’ [64]. Contents 1 Introduction 3 2C∞-rings 5 2.1 Two definitions of C∞-ring . . . . . . . . . . . . . . . . . . . . . 6 2.2C∞-rings as commutative R-algebras, and ideals . . . . . . . . . 7 2.3 Local C∞-rings, and localization . . . . . . . . . . . . . . . . . . 9 2.4 FairC∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Pushouts of C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Flat ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 TheC∞-ringC∞(X)of a manifold X 17 4C∞-ringed spaces and C∞-schemes 20 4.1 Some basic topology . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Sheaves on topological spaces . . . . . . . . . . . . . . . . . . . . 21 4.3C∞-ringed spaces and local C∞-ringed spaces . . . . . . . . . . . 24 14.4 The spectrum functor . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Affine C∞-schemes and C∞-schemes . . . . . . . . . . . . . . . . 31 4.6 Complete C∞-rings . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.8 A criterion for affine C∞-schemes . . . . . . . . . . . . . . . . . . 39 4.9 Quotients of C∞-schemes by finite groups . . . . . . . . . . . . . 42 5 Modules over C∞-rings and C∞-schemes 44 5.1 Modules over C∞-rings . . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Cotangent modules of C∞-rings . . . . . . . . . . . . . . . . . . . 45 5.3 Sheaves ofOX-modules on a C∞-ringed space ( X,OX) . . . . . 50 5.4 Sheaves on affine C∞-schemes, MSpec and Γ . . . . . . . . . . . 51 5.5 Complete modules over C∞-rings . . . . . . . . . . . . . . . . . . 56 5.6 Cotangent sheaves of C∞-schemes . . . . . . . . . . . . . . . . . 58 6C∞-stacks 61 6.1C∞-stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Properties of 1-morphisms of C∞-stacks . . . . . . . . . . . . . . 64 6.3 Open C∞-substacks and open covers . . . . . . . . . . . . . . . . 66 6.4 The underlying topological space of a C∞-stack . . . . . . . . . . 67 6.5 Gluing C∞-stacks by equivalences . . . . . . . . . . . . . . . . . 70 7 Deligne–Mumford C∞-stacks 71 7.1 Quotient C∞-stacks, 1-morphisms, and 2-morphisms . . . . . . . 71 7.2 Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . . . . . . 73 7.3 Characterizing Deligne–Mumford C∞-stacks . . . . . . . . . . . . 76 7.4 Quotient C∞-stacks, 1- and 2-morphisms as local models for objects, 1- and 2-morphisms in DMC∞Sta. . . . . . . . . . . . 80 7.5 Effective Deligne–Mumford C∞-stacks . . . . . . . . . . . . . . . 86 7.6 Orbifolds as Deligne–Mumford C∞-stacks . . . . . . . . . . . . . 87 8 Sheaves on Deligne–Mumford C∞-stacks 89 8.1 Quasicoherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 89 8.2 Writing sheaves in terms of a groupoid presentation . . . . . . . 92 8.3 Pullback of sheaves as a weak 2-functor . . . . . . . . . . . . . . 93 8.4 Cotangent sheaves of Deligne–Mumford C∞-stacks . . . . . . . . 96 9 Orbifold strata of C∞-stacks 99 9.1 The definition of orbifold strata XΓ,...,ˆXΓ ◦. . . . . . . . . . . . 100 9.2 Lifting 1- and 2-morphisms to orbifold strata . . . . . . . . . . . 108 9.3 Orbifold strata of quotient C∞-stacks [X/G] . . . . . . . . . . . 109 9.4 Sheaves on orbifold strata . . . . . . . . . . . . . . . . . . . . . . 111 9.5 Sheaves on orbifold strata of quotients [ X/G] . . . . . . . . . . . 114 9.6 Cotangent sheaves of orbifold strata . . . . . . . . . . . . . . . . 116 2A Background material on stacks 118 A.1 Introduction to 2-categories . . . . . . . . . . . . . . . . . . . . . 118 A.2 Grothendieck topologies, sites, prestacks, and stacks . . . . . . . 122 A.3 Descent theory on a site . . . . . . . . . . . . . . . . . . . . . . . 125 A.4 Properties of 1-morphisms . . . . . . . . . . . . . . . . . . . . . . 126 A.5 Geometric stacks, and stacks associated to groupoids . . . . . . . 128 References 132 Glossary of Notation 137 Index 140 1 Introduction LetXbe a smooth manifold, and write C∞(X) for the set of smooth functions c:X→R. ThenC∞(X) is a commutative R-algebra, with operations of addition, multiplication, and scalar multiplication defined pointwise. How ever, C∞(X) has much more structure than this. For example, if c:X→Ris smooth then exp( c) :X→Ris smooth, and this defines an operation exp : C∞(X)→C∞(X) which cannot be expressed algebraically in terms of the R- algebra structure. More generally, if n/greaterorequalslant0 andf:Rn→Ris smooth, define ann-fold operation Φ f:C∞(X)n→C∞(X) by /parenleftbig Φf(c1,...,cn)/parenrightbig (x) =f/parenleftbig c1(x),...,cn(x)/parenrightbig , for allc1,...,cn∈C∞(X) andx∈X. These operations satisfy many identities: supposem,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→Rare smooth functions. Define a smooth function h:Rn→Rby h(x1,...,xn) =g/parenleftbig f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig , for all (x1,...,xn)∈Rn. Then for all c1,...,cn∈C∞(X) we have Φh(c1,...,cn) = Φg/parenleftbig Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig .(1.1) AC∞-ring/parenleftbig C,(Φf)f:Rn→RC∞/parenrightbig is a setCwith operations Φ f:Cn→Cfor allf:Rn→Rsmooth satisfying identities (1.1), and one other condition. For exampleC∞(X) is aC∞-ring for any manifold X, but there are also many C∞- rings which do not come from manifolds, and can be thought of as rep resenting geometric objects which generalize manifolds. The most basic objects in conventional algebraic geometry are com mutative ringsR, or commutative K-algebrasRfor some field K. The ‘spectrum’ Spec R ofRis an affine scheme, and Ris interpreted as an algebra of functions on SpecR. More general kinds of spaces in algebraic geometry — schemes and stacks — are locally modelled on affine schemes Spec R. This book lays down the foundations of Algebraic Geometry over C∞-rings, in which we replace 3commutative rings in algebraic geometry by C∞-rings. It includes the study of C∞-schemes andDeligne–Mumford C∞-stacks, two classes of geometric spaces generalizing manifolds and orbifolds, respectively. This is not a new idea, but was studied years ago as part of synthetic dif- ferential geometry , which grew out of ideas of Lawvere in the 1960s; see for instance Dubuc [23] on C∞-schemes, and the books by Moerdijk and Reyes [54] and Kock [44]. However, we have new things to say, as we are motivat ed by different problems (see below), and so are asking different question s. Following Dubuc’s discussion of ‘models of synthetic differential geome try’ [21] and oversimplifying a bit, synthetic differential geometers are in terested inC∞-schemes as they provide a category C∞Schof geometric objects which includes smooth manifolds and certain ‘infinitesimal’ objects, and all fi bre prod- ucts exist in C∞Sch, andC∞Schhas some other nice properties to do with open covers, and exponentials of infinitesimals. Synthetic differential geometry concerns proving theorems abou t manifolds usingsyntheticreasoninginvolving‘infinitesimals’. But oneneedstoch eckthese methods of synthetic reasoning are valid. To do this you need a ‘mode l’, some category of geometric spaces including manifolds and infinitesimals, in which you can think of your synthetic arguments as happening. Once you know there exists at least one model with the properties you want, then as far as synthetic differential geometry is concerned the job is done. For this reason C∞-schemes have not been developed very far in synthetic differential geometr y. Recently,C∞-rings andC∞-ringed spaces appeared in a very different con- text, in the theory of derived differential geometry , the differential-geometric analogue of the derived algebraic geometry of Lurie [48] and To¨ en– Vezzosi [66,67], which studies derived smooth manifolds andderived smooth orbifolds . This began with a short section in Lurie [48, §4.5], where he sketched how to define an∞-category of derivedC∞-schemes , including derived manifolds. Lurie’s student David Spivak [64] worked out the details of this, defi ning an∞-category of derived manifolds. Simplifications and extensions of Sp ivak’s theory were given by Borisov and Noel [9,10] and the author [36–38 ]. An al- ternative approach to the foundations of derived differential geo metry involving differential graded C∞-rings is proposed by Carchedi and Roytenberg [12,13]. The author’s notion of derived manifolds [36–38] are called d-manifolds , and are built using our theory of C∞-schemes and quasicoherent sheaves upon them below. They form a 2-category. We also study orbifold versions, d-orbifolds , which are built using our theory of Deligne–Mumford C∞-stacks and their qua- sicoherent sheaves below. Many areas of symplectic geometry involve studying moduli spaces o fJ- holomorphic curves in a symplectic manifold, which are made into Kuranishi spacesin the framework of Fukaya, Oh, Ohta and Ono [26,27]. The author argues that Kuranishi spaces are really derived orbifolds , and has given a new definition [39,41] of a 2-category of Kuranishi spaces Kurwhich is equivalent to the 2-category of d-orbifolds dOrbfrom [36–38]. Because of this, derived differential geometry will have important applications in symplectic ge ometry. To set up our theory of d-manifolds and d-orbifolds requires a lot of pre- 4liminary work on C∞-schemes and C∞-stacks, and quasicoherent sheaves upon them. That is the purpose of this book. We have tried to present a c om- plete, self-contained account which should be understandable to r eaders with a reasonable background in algebraic geometry, and we assume no f amiliarity with synthetic differential geometry. We expect this material may h ave other applications quite different to those the author has in mind in [36–38]. Section 2 explains C∞-rings. The archetypal examples of C∞-rings,C∞(X) for manifolds X, are discussed in §3. Section 4 studies C∞-schemes, and§5 modules over C∞-rings and sheaves of modules over C∞-schemes. Sections 6–9 discuss C∞-stacks. Section 6 defines the 2-category C∞Sta ofC∞-stacks, analogues of Artin stacks in algebraic geometry, and §7 the 2- subcategory DMC∞StaofDeligne–Mumford C∞-stacks, which are C∞-stacks locallymodelled on[ U/G]forUanaffineC∞-schemeand Gafinitegroupacting onU, and are analogues of Deligne–Mumford stacks in algebraic geometr y. We show that orbifolds Orbmay be regarded as a 2-subcategory of DMC∞Sta. Section 8 studies quasicoherent sheaves on Deligne–Mumford C∞-stacks, gen- eralizing§5, and§9 orbifold strata of Deligne–Mumford C∞-stacks. Appendix Asummarizesbackgroundonstacksfrom[3,4,29,46,49 ,55], foruse in§6–§9. Stacks are a very technical area, and §A is too terse to help a beginner learn the subject, it is intended only to establish notation and definit ions for those already familiar with stacks. Readers with no experience of st acks are advised to first consult an introductory text such as Vistoli [68], G omez [29], Laumon and Moret-Bailly [46], or the online ‘Stacks Project’ [34]. Much of§2–§4 is already understood in synthetic differential geometry, such as in the work of Dubuc [23] and Moerdijk and Reyes [54]. But we believ e it is worthwhile giving a detailed and self-contained exposition, from our o wn point of view. Sections 5–9 are new, so far as the author knows, though §5–§8 are based on well known material in algebraic geometry. Acknowledgements. I would like to thank Omar Antolin, Eduardo Dubuc, Kelli Francis-Staite, Jacob Gross, Jacob Lurie, and Ieke Moerdijk for helpful conver- sations, and a referee for many useful comments. This research was supported by EPSRC grants EP/H035303/1 and EP/J016950/1. 2C∞-rings We begin by explaining the basic objects out of which our theories are built, C∞-rings, orsmooth rings . The archetypal example of a C∞-ring is the vector spaceC∞(X) of smooth functions c:X→Rfor a manifold X. Everything in thissectionis knowntoexpertsin syntheticdifferentialgeometry, andmuchofit canbe found in Moerdijkand Reyes[54, Ch. I], Dubuc [21–24] orKock [44,§III]. We introducesomenew notationfor brevity, in particular, our fairC∞-ringsare known in the literature as ‘finitely generated and germ determined C∞-rings’. 52.1 Two definitions of C∞-ring We first define C∞-rings in the style of classical algebra. Definition 2.1. AC∞-ringis a setCtogether with operations Φf:Cn=/rightanglenwncopies/rightanglene C×···×C−→C for alln/greaterorequalslant0 and smooth maps f:Rn→R, where by convention when n= 0 we defineC0to be the single point {∅}. These operations must satisfy the following relations: suppose m,n/greaterorequalslant0, andfi:Rn→Rfori= 1,...,mandg:Rm→R are smooth functions. Define a smooth function h:Rn→Rby h(x1,...,xn) =g/parenleftbig f1(x1,...,xn),...,fm(x1...,xn)/parenrightbig , for all (x1,...,xn)∈Rn. Then for all ( c1,...,cn)∈Cnwe have Φh(c1,...,cn) = Φg/parenleftbig Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig . We also require that for all 1 /lessorequalslantj/lessorequalslantn, definingπj:Rn→Rbyπj: (x1,...,xn)/ma√sto→xj, we have Φ πj(c1,...,cn) =cjfor all (c1,...,cn)∈Cn. Usually we refer to Cas theC∞-ring, leaving the operations Φ fimplicit. Amorphism betweenC∞-rings/parenleftbig C,(Φf)f:Rn→RC∞/parenrightbig ,/parenleftbig D,(Ψf)f:Rn→RC∞/parenrightbig is a mapφ:C→Dsuch that Ψ f/parenleftbig φ(c1),...,φ(cn)/parenrightbig =φ◦Φf(c1,...,cn) for all smooth f:Rn→Randc1,...,cn∈C. We will write C∞Ringsfor the category of C∞-rings. Here is the motivating example, which we will study at greater length in §3: Example 2.2. LetXbe a manifold, which may be without boundary, or with boundary, or with corners. Write C∞(X) for the set of smooth functions c: X→R. Forn/greaterorequalslant0 andf:Rn→Rsmooth, define Φ f:C∞(X)n→C∞(X) by /parenleftbig Φf(c1,...,cn)/parenrightbig (x) =f/parenleftbig c1(x),...,cn(x)/parenrightbig , (2.1) for allc1,...,cn∈C∞(X) andx∈X. It is easy to see that C∞(X) and the operations Φ fform aC∞-ring. Example 2.3. TakeXto be the point∗in Example 2.2. Then C∞(∗) =R, with operations Φ f:Rn→Rgiven by Φ f(x1,...,xn) =f(x1,...,xn). This makesRinto the simplest nonzero example of a C∞-ring, the initial object inC∞Rings. Note thatC∞-rings are far more general than those coming from manifolds. For example, if Xis any topological space we could define a C∞-ringC0(X) to be the set of continuous c:X→Rwith operations Φ fdefined as in (2.1). For Xa manifold with dim X >0, theC∞-ringsC∞(X) andC0(X) are different. There is a more succinct definition of C∞-rings using category theory: 6Definition 2.4. WriteManfor the category of manifolds, and Eucfor the full subcategory of Manwith objects the Euclidean spaces Rn. That is, the objects ofEucareRnforn= 0,1,2,...,and the morphisms in Eucare smooth maps f:Rm→Rn. Write Setsfor the category of sets. In both EucandSets we have notions of (finite) products of objects (that is, Rm+n=Rm×Rn, and productsS×Tof setsS,T), and products of morphisms. Define a ( category-theoretic )C∞-ringto be a product-preserving functor F:Euc→Sets. HereFshould also preserve the empty product, that is, it mapsR0inEucto the terminal object in Sets, the point∗. C∞-rings in this sense are an example of an algebraic theory in the sense of Ad´ amek, Rosick´ y and Vitale [1], and many of the basic categorical p roperties ofC∞-rings follow from this. Here is how this relates to Definition 2.1. Suppose F:Euc→Setsis a product-preserving functor. Define C=F(R). Then Cis an object in Sets, that is, a set. Suppose n/greaterorequalslant0 andf:Rn→Ris smooth. Then fis a morphism inEuc, soF(f) :F(Rn)→F(R) =Cis a morphism in Sets. SinceFpreserves productsF(Rn) =F(R)×···×F(R) =Cn, soF(f) mapsCn→C. We define Φf:Cn→Cby Φf=F(f). The fact that Fis a functor implies that the Φ f satisfy the relations in Definition 2.1, so/parenleftbig C,(Φf)f:Rn→RC∞/parenrightbig is aC∞ring. Conversely, if/parenleftbig C,(Φf)f:Rn→RC∞/parenrightbig is aC∞-ring then we define F:Euc→ SetsbyF(Rn) =Cn, and iff:Rn→Rmis smooth then f= (f1,...,fm) for fi:Rn→Rsmooth, and we define F(f) :Cn→CmbyF(f) : (c1,...,cn)/ma√sto→/parenleftbig Φf1(c1,...,cn),...,Φfm(c1,...,cn)/parenrightbig . ThenFis a product-preserving functor. This defines a 1-1 correspondence between C∞-rings in the sense of Definition 2.1, and category-theoretic C∞-rings in the sense of Definition 2.4. As in Moerdijk and Reyes [54, p. 21–22] we have: Proposition 2.5. In the category C∞RingsofC∞-rings, all limits and all filtered colimits exist, and regarding C∞-rings as functors F:Euc→Sets as in Definition 2.4,they may be computed objectwise in Eucby taking the corresponding limits/filtered colimits in Sets. Also, all small colimits exist, though in general they are no t computed ob- jectwise in Eucby taking colimits in Sets. In particular, pushouts and all finite colimits exist in C∞Rings. We will write D∐φ,C,ψEorD∐CEfor the pushout of morphisms φ:C→D, ψ:C→EinC∞Rings. WhenC=R, the initial object in C∞Rings, pushouts D∐REare called coproducts and are usually written D⊗∞E. ForR-algebras A,Bthe coproduct is the tensor product A⊗B. But the coproduct D⊗∞Eof C∞-ringsD,Eis generally different from their coproduct D⊗EasR-algebras. For example we have C∞(Rm)⊗∞C∞(Rn)∼=C∞(Rm+n), which contains but is much larger than the tensor product C∞(Rm)⊗C∞(Rn) form,n>0. 2.2C∞-rings as commutative R-algebras, and ideals EveryC∞-ringChas an underlying commutative R-algebra: 7Definition 2.6. LetCbe aC∞-ring. Then we may give Cthe structure of acommutative R-algebra. Define addition ‘+’ on Cbyc+c′= Φf(c,c′) for c,c′∈C, wheref:R2→Risf(x,y) =x+y. Define multiplication ‘ ·’ onCby c·c′= Φg(c,c′), whereg:R2→Risf(x,y) =xy. Define scalar multiplication byλ∈Rbyλc= Φλ′(c), whereλ′:R→Risλ′(x) =λx. Define elements 0 and 1 in Cby 0 = Φ 0′(∅) and 1 = Φ 1′(∅), where 0′:R0→Rand 1′:R0→R are the maps 0′:∅/ma√sto→0 and 1′:∅/ma√sto→1. The relations on the Φ fimply that all the axioms of a commutative R-algebra are satisfied. In Example 2.2, this yields the obvious R-algebra structure on the smooth functions c:X→R. Here is another way to say this. In an R-algebraA, then-fold ‘operations’ Φ :An→A, that is, all the maps An→Awe can construct using only addition, multiplication, scalar multiplication, and the elements 0 ,1∈A, correspond ex- actly to polynomials p:Rn→R. Since polynomials are smooth, the operations of anR-algebra are a subset of those of a C∞-ring, and we can truncate from C∞-rings to R-algebras. As there are many more smooth functions f:Rn→R than there are polynomials, a C∞-ring has far more structure and operations than a commutative R-algebra. Definition 2.7. AnidealIinCis an idealI⊂CinCregarded as a commu- tativeR-algebra. Then we make the quotient C/Iinto aC∞-ring as follows. If f:Rn→Ris smooth, define ΦI f: (C/I)n→C/Iby ΦI f(c1+I,...,cn+I) = Φf(c1,...,cn)+I. To show this is well-defined, we must show it is independent of the choic e of representatives c1,...,cninCforc1+I,...,cn+IinC/I. By Hadamard’s Lemma there exist smooth functions gi:R2n→Rfori= 1,...,nwith f(y1,...,yn)−f(x1,...,xn) =/summationtextn i=1(yi−xi)gi(x1,...,xn,y1,...,yn) for allx1,...,xn,y1,...,yn∈R. Ifc′ 1,...,c′ nare alternative choices for c1,..., cn, so thatc′ i+I=ci+Ifori= 1,...,nandc′ i−ci∈I, we have Φf(c′ 1,...,c′ n)−Φf(c1,...,cn) =/summationtextn i=1(c′ i−ci)Φgi(c′ 1,...,c′ n,c1,...,cn). The second line lies in Iasc′ i−ci∈IandIis an ideal, so ΦI fis well-defined, and clearly/parenleftbig C/I,(ΦI f)f:Rn→RC∞/parenrightbig is aC∞-ring. IfCis aC∞-ring, we will use the notation ( fa:a∈A) to denote the ideal inCgenerated by a collection of elements fa,a∈AinC, in the sense of commutative R-algebras. That is, (fa:a∈A) =/braceleftbig/summationtextn i=1fai·ci:n/greaterorequalslant0,a1,...,an∈A,c1,...,cn∈C/bracerightbig . Definition 2.8. AC∞-ringCis calledfinitely generated if there exist c1,...,cn inCwhich generate Cover allC∞-operations. That is, for each c∈Cthere exists a smooth map f:Rn→Rwithc= Φf(c1,...,cn). (This is a much weaker condition than Cbeing finitely generated as a commutative R-algebra.) 8By Kock [44, Prop. III.5.1], C∞(Rn) is the free C∞-ring withngenerators. Given such C,c1,...,cn, defineφ:C∞(Rn)→Cbyφ(f) = Φf(c1,...,cn) for smoothf:Rn→R, whereC∞(Rn) is as in Example 2.2 with X=Rn. Then φis a surjective morphism of C∞-rings, soI= Kerφis an ideal in C∞(Rn), andC∼=C∞(Rn)/Ias aC∞-ring. Thus, Cis finitely generated if and only if C∼=C∞(Rn)/Ifor somen/greaterorequalslant0 and ideal IinC∞(Rn). AnidealIinaC∞-ringCiscalledfinitelygenerated ifIisafinitelygenerated ideal of the underlying commutative R-algebra of Cin Definition 2.6, that is, I= (i1,...,ik)forsomei1,...,ik∈C. AC∞-ringCiscalledfinitelypresented if C∼=C∞(Rn)/Ifor somen/greaterorequalslant0, whereIis a finitely generated ideal in C∞(Rn). A difference with conventional algebraic geometry is that C∞(Rn) is not noetherian, so ideals in C∞(Rn) may not be finitely generated, and Cfinitely generated does not imply Cfinitely presented. WriteC∞RingsfgandC∞Ringsfpfor the full subcategories of finitely generated and finitely presented C∞-rings in C∞Rings. Example 2.9. AWeil algebra [21, Def. 1.4]is afinite-dimensional commutative R-algebraWwhich has a maximal ideal mwithW/m∼=Randmn= 0 for some n >0. Then by Dubuc [21, Prop. 1.5] or Kock [44, Th. III.5.3], there is a unique way to make Winto aC∞-ring compatible with the given underlying commutative R-algebra. This C∞-ring is finitely presented [44, Prop. III.5.11]. C∞-rings from Weil algebras are important in synthetic differential geo metry, in arguments involving infinitesimals. See [11, §2] for a detailed study of this. 2.3 Local C∞-rings, and localization Definition 2.10. AC∞-ringCis called localif regarded as an R-algebra, as in Definition 2.6, Cis a local R-algebra with residue field R. That is, Chas a unique maximal ideal mCwithC/mC∼=R. IfC,Dare localC∞-rings with maximal ideals mC,mD, andφ:C→Dis a morphism of C∞rings, then using the fact that C/mC∼=R∼=D/mDwe see thatφ−1(mD) =mC, that is,φis alocalmorphism of local C∞-rings. Thus, there is no difference between morphisms and local morphisms. Remark 2.11. We use the term ‘local C∞-ring’ following Dubuc [23, Def. 4]. They are also called C∞-local rings in Dubuc [22, Def. 2.13], pointed local C∞- ringsin [54,§I.3] andArchimedean local C∞-ringsin [52,§3]. Moerdijk and Reyes [52–54] use the term ‘local C∞-ring’ to mean a C∞-ring which is a local R-algebra, but which need not have residue field R. The next example is taken from Moerdijk and Reyes [54, §I.3]. Example 2.12. WriteC∞(N) for theR-algebraofallfunctions f:N→R. It is a finitely generated C∞-ring isomorphic to C∞(R)/{f∈C∞(R) :f|N= 0}. Let Fbe anon-principal ultrafilter onN, in the sense of Comfort and Negrepontis [16], and let I⊂Cbe the prime ideal of f:N→Rsuch that{n∈N:f(n) = 0} lies inF. ThenC=C∞(N)/Iis a finitely generated C∞-ring which is a local 9R-algebra by [54, Ex. I.3.2], that is, it has a unique maximal ideal mC, but its residue field is not Rby [54, Cor. I.3.4]. Hence Cis a localC∞-ring in the sense of [52–54], but not in our sense. Localizations ofC∞-rings are studied in [22,23,52,53], see [54, p. 23]. Definition 2.13. LetCbe aC∞-ring andSa subset of C. Alocalization C[s−1:s∈S] ofCatSis aC∞-ringD=C[s−1:s∈S] and a morphism π:C→Dsuch thatπ(s) is invertible in Dfor alls∈S, with the universal property that if Eis aC∞-ring andφ:C→Ea morphism with φ(s) invertible inEfor alls∈S, then there is a unique morphism ψ:D→Ewithφ=ψ◦π. A localization C[s−1:s∈S] always exists — it can be constructed by adjoining an extra generator s−1and an extra relation s·s−1−1 = 0 for each s∈S— and is unique up to unique isomorphism. When S={c}we have an exact sequence 0 →I→C⊗∞C∞(R)π−→C[c−1]→0, where C⊗∞C∞(R) is the coproduct of C,C∞(R) as in§2.1, with pushout morphisms ι1:C→ C⊗∞C∞(R),ι2:C∞(R)→C⊗∞C∞(R), andIis the ideal in C⊗∞C∞(R) generated by ι1(c)·ι2(x)−1, wherexis the generator of C∞(R). AnR-pointxof aC∞-ringCis aC∞-ring morphism x:C→R, where Ris regarded as a C∞-ring as in Example 2.3. By [54, Prop. I.3.6], a map x:C→Ris a morphism of C∞-rings if and only if it is a morphism of the underlying R-algebras, as in Definition 2.6. Define Cxto be the localization Cx=C[s−1:s∈C,x(s)/\e}atio\slash= 0], with projection πx:C→Cx. ThenCxis a localC∞-ring by [53, Lem. 1.1]. The R-points ofC∞(Rn) are just evaluation at pointsx∈Rn. This also holds for C∞(X) for any manifold X. In a new result, we can describe these local C∞-ringsCxexplicitly. Note that the surjectivity of πx:C→Cxin the next proposition is surprising. It doesnot hold forgenerallocalizationsof C∞-rings— forinstance, π:C∞(R)→ C∞(R)[x−1] is injective but not surjective, as x−1/∈Imπ— or for localizations πx:A→Axof rings or K-algebras in conventional algebraic geometry. Proposition 2.14. LetCbe aC∞-ring,x:C→RanR-point of C,andCx the localization, with projection πx:C→Cx. Thenπxis surjective with kernel an idealI⊂C,so thatCx∼=C/I,where I=/braceleftbig c∈C:there exists d∈Cwithx(d)/\e}atio\slash= 0inRandc·d= 0inC/bracerightbig .(2.2) Proof.ClearlyIin (2.2) is closed under multiplication by elements of C. Let c1,c2∈I, so there exist d1,d2∈Cwithx(d1)/\e}atio\slash= 0/\e}atio\slash=x(d2) andc1d1= 0 =c2d2. Thend1d2∈Cwithx(d1d2) =x(d1)x(d2)/\e}atio\slash= 0, and(c1+c2)(d1·d2) =d2(c1d1)+ d1(c2d2) = 0, soc1+c2∈I. HenceIis an ideal, and C/IaC∞-ring. Supposec∈I, so there exists d∈Cwithx(d)/\e}atio\slash= 0 andcd= 0. Then πx(d) is invertible in Cxby definition. Thus πx(c) =πx(c)πx(d)πx(d)−1=πx(cd)πx(d)−1=πx(0)πx(d)−1= 0. ThereforeI⊆Kerπx. Soπx:C→Cxfactorizes uniquely as πx=ı◦π, where π:C→C/Iis the projection and ı:C/I→Cxis aC∞-ring morphism. 10Supposec∈Cwithx(c)/\e}atio\slash= 0, and write ǫ=1 2|x(c)|. Choose smooth functionsη:R→R\{0}, so thatη−1:R→R\{0}is also smooth, such that η(t) =tfor allt∈(x(c)−ǫ,x(c)+ǫ), andζ:R→Rsuch thatζ(t) = 0 for all t∈R\(x(c)−ǫ,x(c)+ǫ), so that (η−idR)·ζ= 0, andζ(x(c)) = 1. Setc1= Φη(c),c2= Φη−1(c) andd= Φζ(c) inC, using the C∞-ring operations from η,η−1,ζ. Thenc1c2= 1 inC, asη·η−1= 1, andx(d) = x(Φζ(c)) =ζ(x(c)) = 1, asx:C→Ris aC∞-ring morphism. Also (c1−c)·d=/parenleftbig Φη(c)−ΦidR(c)/parenrightbig Φζ(c) = Φ(η−idR)ζ(c) = Φ0(c) = 0. Hencec1−c∈Iasx(d)/\e}atio\slash= 0, soc+I=c1+I. But then ( c+I)(c2+I) = (c1+I)(c2+I) =c1c2+I= 1+IinC/I, soπ(c) =c+Iis invertible in C/I. As this holds for all c∈Cwithx(c)/\e}atio\slash= 0, by the universal property of Cx there exists a unique C∞-ring morphism :Cx→C/Iwithπ=◦πx. Since πx,πare surjective, πx=ı◦πandπ=◦πximply that ı:C/I→Cxand :Cx→C/Iare inverse, so both are isomorphisms. Example 2.15. Forn/greaterorequalslant0 andp∈Rn, defineC∞ p(Rn) to be the set of germs of smooth functions c:Rn→Ratp∈Rn, made into a C∞-ring in the obvious way. Then C∞ p(Rn) is a localC∞-ring in the sense of Definition 2.10. Here are three different ways to define C∞ p(Rn), which yield isomorphic C∞-rings: (a) Defining C∞ p(Rn) asthe germsoffunctions ofsmoothfunctionsat pmeans that points of C∞ p(Rn) are∼-equivalence classes [( U,c)] of pairs ( U,c), whereU⊆Rnis open with p∈Uandc:U→Ris smooth, and (U,c)∼(U′,c′) if there exists p∈V⊆U∩U′open withc|V≡c′|V. (b) As the localization ( C∞(Rn))p=C∞(Rn)[g∈C∞(Rn) :g(p)/\e}atio\slash= 0]. Then points of (C∞(Rn))pare equivalence classes [ f/g] of fractions f/gfor f,g∈C∞(Rn) withg(p)/\e}atio\slash= 0, and fractions f/g,f′/g′are equivalent if there exists h∈C∞(Rn) withh(p)/\e}atio\slash= 0 andh(fg′−f′g)≡0. (c) As the quotient C∞(Rn)/I, whereIis the ideal of f∈C∞(Rn) with f≡0 nearp∈Rn. One can show (a)–(c) are isomorphic using the fact that if Uis any open neigh- bourhood of pinRnthen there exists smooth η:Rn→[0,1] such that η≡0 on an open neighbourhood of Rn\UinRnandη≡1 on an open neighbourhood ofpinU. By Moerdijk and Reyes [54, Prop. I.3.9], any finitely generated local C∞-ring is a quotient of some C∞ p(Rn). 2.4 Fair C∞-rings We now discuss an important class of C∞-rings, which we call fairC∞-rings, for brevity. Although our term ‘fair’ is new, we stress that the idea is already well-known, being originally introduced by Dubuc [22], [23, Def. 11], who first recognized their significance, under the name ‘ C∞-rings of finite type presented by an ideal of local character’, and in more recent works would be re ferred to as ‘finitely generated and germ-determined C∞-rings’. 11Definition 2.16. An idealIinC∞(Rn) is called fairif for eachf∈C∞(Rn), flies inIif and only if πp(f) lies inπp(I)⊆C∞ p(Rn) for allp∈Rn, where C∞ p(Rn) is as in Example 2.15 and πp:C∞(Rn)→C∞ p(Rn) is the natural projectionπp:c/ma√sto→[(Rn,c)]. AC∞-ringCis called fairif it is isomorphic toC∞(Rn)/I, whereIis a fair ideal. Equivalently, Cis fair if it is finitely generated and whenever c∈Cwithπp(c) = 0 in Cpfor allR-pointsp:C→R thenc= 0, using the notation of Definition 2.13. Dubuc [22], [23, Def. 11] calls fair ideals ideals of local character , and Mo- erdijk and Reyes [54, I.4] call them germ determined , which has now become the accepted term. Fair C∞-rings are also sometimes called germ determined C∞- rings, a more descriptive term than ‘fair’, but the definition of germ deter mined C∞-ringsCin [54, Def. I.4.1] does not require Cfinitely generated, so does not equate exactly to our fair C∞-rings. By Dubuc [22, Prop. 1.8], [23, Prop. 12] any finitely generated ideal Iis fair, so Cfinitely presented implies Cfair. We writeC∞Ringsfafor the full subcategory of fair C∞-rings in C∞Rings. Proposition 2.17. SupposeI⊂C∞(Rm)andJ⊂C∞(Rn)are ideals with C∞(Rm)/I∼=C∞(Rn)/JasC∞-rings. Then Iis finitely generated, or fair, if and only if Jis finitely generated, or fair, respectively. Proof.Writeφ:C∞(Rm)/I→C∞(Rn)/Jfor the isomorphism, and x1,...,xm for the generators of C∞(Rm), andy1,...,ynfor the generators of C∞(Rn). Sinceφis an isomorphism we can choose f1,...,fm∈C∞(Rn) withφ(xi+I) = fi+Jfori= 1,...,mandg1,...,gn∈C∞(Rm) withφ(gi+I) =yi+Jfor i= 1,...,n. It is now easy to show that I=/parenleftbig xi−fi/parenleftbig g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig , i= 1,...,m, andh/parenleftbig g1(x1,...,xm),...,gn(x1,...,xm)/parenrightbig , h∈J/parenrightbig . Hence, ifJisgeneratedby h1,...,hkthenIisgeneratedby xi−fi(g1,...,gn) fori= 1,...,mandhj(g1,...,gn) forj= 1,...,k, soJfinitely generated impliesIfinitelygenerated. Applyingthesameargumentto φ−1:C∞(Rn)/J→ C∞(Rm)/I, we see that Iis finitely generated if and only if Jis. SupposeIis fair, and let f∈C∞(Rn) withπq(f)∈πq(J)⊆C∞ q(Rn) for allq∈Rn. We will show that f∈J, so thatJis fair. Consider the function f′=f(g1,...,gn)∈C∞(Rm). Ifp= (p1,...,pm) inRmandq= (q1,...,qn) =/parenleftbig g1(p1,...,pm),...,gn(p1,...,pm)/parenrightbig thenφ:C∞(Rm)/I→C∞(Rn)/Jlocalizes to an isomorphism φp:C∞ p(Rm)/πp(I)→C∞ q(Rn)/πq(J) which maps φp: πp(f′)+πp(I)/ma√sto→πq(f)+πq(J). Sinceπq(f)∈πq(J), this gives πp(f′)∈πp(I) for allp∈Rm, sof′∈IasIis fair. But φ(f′+I) =f+J, sof′∈Iimplies f∈J. Therefore Jis fair. Conversely, Jis fair implies Iis fair. Example 2.18. The localC∞-ringC∞ p(Rn) of Example 2.15 is the quotient of C∞(Rn) by the ideal Iof functions fwithf≡0 nearp∈Rn. Forn>0 thisI is fair, but not finitely generated. So C∞ p(Rn) is fair, but not finitely presented, by Proposition 2.17. 12The following example taken from Dubuc [24, Ex. 7.2] shows that localiz a- tions of fair C∞-rings need not be fair: Example 2.19. LetCbe the local C∞-ringC∞ 0(R), as in Example 2.15. Then C∼=C∞(R)/I, whereIis the ideal of all f∈C∞(R) withf≡0 near 0 in R. ThisIis fair, so Cis fair. Let c= [(x,R)]∈C. Then the localization C[c−1] is theC∞-ring of germs at 0 in Rof smooth functions R\{0}→R. Taking y=x−1as a generator of C[c−1], we see that C[c−1]∼=C∞(R)/J, whereJis the ideal of compactly supported functions in C∞(R). ThisJis not fair, so by Proposition 2.17, C[c−1] is not fair. Recall from category theory that if Cis a subcategory of a category D, a reflectionR:D→Cisaleft adjointtothe inclusion C֒→D. Thatis,R:D→C is a functor with natural isomorphisms Hom C(R(D),C)∼=HomD(D,C) for all C∈CandD∈D. We will define a reflection for C∞Ringsfa⊂C∞Ringsfg, following Moerdijk and Reyes [54, p. 48–49] (see also Dubuc [23, Th. 13]). Definition 2.20. LetCbe a finitely generated C∞-ring. LetICbe the ideal of allc∈Csuch thatπp(c) = 0 in Cpfor allR-pointsp:C→R. ThenC/IC is a finitely generated C∞-ring, with projection π:C→C/IC. It has the same R-points as C, that is, morphisms p:C/IC→Rare in 1-1 correspondence with morphisms p′:C→Rbyp′=p◦π, and the local rings ( C/IC)pandCp′ are naturally isomorphic. It follows that C/ICis fair. Define a functor Rfa fg: C∞Ringsfg→C∞RingsfabyRfa fg(C) =C/ICon objects, and if φ:C→D is a morphism then φ(IC)⊆ID, soφinduces a morphism φ∗:C/IC→D/ID, and we set Rfa fg(φ) =φ∗. It is easy to see Rfa fgis a reflection. IfIis an ideal in C∞(Rn), write¯Ifor the set of f∈C∞(Rn) withπp(f)∈ πp(I) for allp∈Rn. Then¯Iis the smallest fair ideal in C∞(Rn) containing I, thegerm-determined closure ofI, andRfa fg/parenleftbig C∞(Rn)/I/parenrightbig∼=C∞(Rn)/¯I. Example 2.21. Letη:R→[0,∞) be smooth with η(x)>0 forx∈(0,1) and η(x) = 0 forx /∈(0,1). DefineI⊆C∞(R) by I=/braceleftbig/summationtext a∈Aga(x)η(x−a) :A⊂Zis finite,ga∈C∞(R),a∈A/bracerightbig . ThenIis an ideal in C∞(R), soC=C∞(R)/Iis aC∞-ring. The set of f∈C∞(R) such that πp(f) lies inπp(I)⊆C∞ p(R) for allp∈Ris ¯I=/braceleftbig/summationtext a∈Zga(x)η(x−a) :ga∈C∞(R), a∈Z/bracerightbig , where the sum/summationtext a∈Zga(x)η(x−a) makes sense as at most one term is nonzero at any point x∈R. Since¯I/\e}atio\slash=I, we see that Iisnot fair, soC=C∞(R)/Iis not a fairC∞-ring. In fact ¯Iis the smallest fair ideal containing I. We have IC∞(R)/I=¯I/I, andRfa fg/parenleftbig C∞(R)/I) =C∞(R)/¯I. Proposition 2.22. LetCbe aC∞-ring, andGa finite group acting on Cby automorphisms. Then the fixed subset CGofGinChas the structure of a C∞- ring in a unique way, such that the inclusion CG֒→Cis aC∞-ring morphism. IfCis fair, or finitely presented, then CGis also fair, or finitely presented. 13Proof.For the first part, let f:Rn→Rbe smooth, and c1,...,cn∈CG. Then γ·Φf(c1,...,cn) = Φf(γ·c1,...,γ·cn) = Φf(c1,...,cn) for eachγ∈G, so Φf(c1,...,cn)∈CG. Define ΦG f: (CG)n→CGby ΦG f= Φf|(CG)n. It is now trivial to check that the operations ΦG ffor smooth f:Rn→RmakeCGinto a C∞-ring, uniquely such that CG֒→Cis aC∞-ring morphism. Suppose now that Cis finitely generated. Choose a finite set of generators forC, and by adding the images of these generators under G, extend to a set of (not necessarily distinct) generators x1,...,xnforC, on whichGacts freely by permutation. This gives an exact sequence 0 ֒→I→C∞(Rn)→C→0, whereC∞(Rn) is freely generated by x1,...,xn. HereRnis a direct sum of copies of the regular representation of G, andC∞(Rn)→CisG-equivariant. HenceIis aG-invariant ideal in C∞(Rn), which is fair, or finitely generated, respectively. Taking G-invariant parts gives an exact sequence 0 ֒→IG→ C∞(Rn)Gπ−→CG→0, whereC∞(Rn)G,CGare clearlyC∞-rings. AsGacts linearly on Rnit acts by automorphisms on the polynomial ring R[x1,...,xn]. By a classical theorem of Hilbert [70, p. 274], R[x1,...,xn]G is a finitely presented R-algebra, so we can choose generators p1,...,plfor R[x1,...,xn]G, which induce a surjective R-algebra morphism R[p1,...,pl]→ R[x1,...,xn]Gwith kernel generated by q1,...,qm∈R[p1,...,pl]. By results of Bierstone [6] for Ga finite group and Schwarz [63] for Ga compact Lie group, any G-invariant smooth function on Rnmay be written as a smooth function of the generators p1,...,plofR[x1,...,xn]G, giving a surjective morphism C∞(Rl)→C∞(Rn)G, whose kernel is the ideal in C∞(Rl) generated by q1,...,qm. ThusC∞(Rn)Gis finitely presented. AlsoCGis generated by π(p1),...,π(pl), soCGis finitely generated, and we have an exact sequence 0 ֒→J→C∞(Rl)π−→CG→0, whereJis the ideal in C∞(Rl) generated by q1,...,qmand the lifts to C∞(Rl) of a generating set for the idealIGinC∞(Rn)G∼=C∞(Rl)/(q1,...,qm). Suppose now that Iis fair. Then for f∈C∞(Rn)G,flies inIGif and only ifπp(f)∈πp(I)⊆C∞ p(Rn) for allp∈Rn. IfHis the subgroup of Gfixing pthenHacts onC∞ p(Rn), andπp(f) isH-invariant as fisG-invariant, and πp(I)H=πp(IG). Thus we may rewrite the condition as flies inIGif and only ifπp(f)∈πp(IG)⊆C∞ p(Rn) for allp∈Rn. Projecting from RntoRn/G, this says thatflies inIGif and only if πp(f) lies inπp(IG)⊆/parenleftbig C∞(Rn)G/parenrightbig pfor all p∈Rn/G. SinceC∞(Rn)Gis finitely presented, it follows as in [54, Cor. I.4.9] thatJis fair, so CGis fair. SupposeIis finitely generated in C∞(Rn), with generators f1,...,fk. As Rnis a sum of copies of the regular representation of G, so that every irre- ducible representation of Goccurs as a summand of Rn, one can show that IG is generated as an ideal in C∞(Rn/G) by then(k+1) elements fG iand (fixj)G fori= 1,...,kandj= 1,...,n, wherefG=1 |G|/summationtext γ∈Gf◦γis theG-invariant part off∈C∞(Rn). Therefore Jis finitely generated by q1,...,qmand lifts of fG i,(fixj)G. Hence if Cis finitely presented then CGis finitely presented. 142.5 Pushouts of C∞-rings Proposition 2.5 shows that pushouts of C∞-rings exist. For finitely generated C∞-rings, we can describe these pushouts explicitly. Example 2.23. Suppose the following is a pushout diagram of C∞-rings: Cβ/d47/d47 α/d15/d15E δ/d15/d15 Dγ/d47/d47F, so thatF=D∐CE, withC,D,Efinitely generated. Then we have exact sequences 0→I ֒→C∞(Rl)φ−→C→0,0→J ֒→C∞(Rm)ψ−→D→0, and 0→K ֒→C∞(Rn)χ−→E→0,(2.3) whereφ,ψ,χare morphisms of C∞-rings, and I,J,Kare ideals in C∞(Rl), C∞(Rm),C∞(Rn). Writex1,...,xlandy1,...,ymandz1,...,znfor the gen- erators ofC∞(Rl),C∞(Rm),C∞(Rn) respectively. Then φ(x1),...,φ(xl) gen- erateC, andα◦φ(x1),...,α◦φ(xl) lie inD, so we may write α◦φ(xi) =ψ(fi) fori= 1,...,lasψis surjective, where fi:Rm→Ris smooth. Similarly β◦φ(x1),...,β◦φ(xl) lie inE, so we may write β◦φ(xi) =χ(gi) fori= 1,...,l, wheregi:Rn→Ris smooth. Then from the explicit construction of pushouts of C∞-rings we obtain an exact sequence with ξa morphism of C∞-rings 0 /d47/d47L /d47/d47C∞(Rm+n)ξ/d47/d47F /d47/d470, (2.4) where we write the generators of C∞(Rm+n) asy1,...,ym,z1,...,zn, and then Lis the ideal in C∞(Rm+n) generated by the elements d(y1,...,ym) ford∈ J⊆C∞(Rm), ande(z1,...,zn) fore∈K⊆C∞(Rn), andfi(y1,...,ym)− gi(z1,...,zn) fori= 1,...,l. For the case of coproducts D⊗∞E, withC=R,l= 0 andI={0}, we have /parenleftbig C∞(Rm)/J/parenrightbig ⊗∞/parenleftbig C∞(Rn)/K/parenrightbig∼=C∞(Rm+n)/(J,K). Proposition 2.24. The subcategories C∞RingsfgandC∞Ringsfpare closed under pushouts and all finite colimits in C∞Rings. Proof.Firstweshow C∞Ringsfg,C∞Ringsfpareclosedunderpushouts. Sup- poseC,D,Eare finitely generated, and use the notation of Example 2.23. Then Fis finitely generated with generators y1,...,ym,z1,...,zn, soC∞Ringsfg is closed under pushouts. If C,D,Eare finitely presented then we can take J= (d1,...,dj) andK= (e1,...,ek), and then Example 2.23 gives L=/parenleftbig dp(y1,...,ym), p= 1,...,j, ep(z1,...,zn), p= 1,...,k, fp(y1,...,ym)−gp(z1,...,zn), p= 1,...,l/parenrightbig .(2.5) 15SoLis finitely generated, and F∼=C∞(Rm+n)/Lis finitely presented. Thus C∞Ringsfpis closed under pushouts. NowRis an initial object in C∞Ringsfg,C∞Ringsfp⊂C∞Rings, and all finite colimits may be constructed by repeated pushouts involving the initial object. Hence C∞Ringsfg,C∞Ringsfpare closed under finite colimits. Here is an example from Dubuc [24, Ex. 7.1], Moerdijk and Reyes [54, p . 49]. Example 2.25. Consider the coproduct C∞(R)⊗∞C∞ 0(R), whereC∞ 0(R) is theC∞-ring of germs of smooth functions at 0 in Ras in Example 2.15. Then C∞(R),C∞ 0(R) are fairC∞-rings, but C∞ 0(R) is not finitely presented. By Example 2.23, C∞(R)⊗∞C∞ 0(R) =C∞(R)∐RC∞ 0(R)∼=C∞(R2)/L, whereL is the ideal in C∞(R2) generated by functions f(x,y) =g(y) forg∈C∞(R) withg≡0 near 0∈R. This ideal Lis not fair, since for example one can findf∈C∞(R2) withf(x,y) = 0 if and only if |xy|/lessorequalslant1, and then f /∈Lbut πp(f)∈πp(L)⊆C∞ p(R2) for allp∈R2. HenceC∞(R)⊗∞C∞ 0(R) is not a fair C∞-ring, by Proposition 2.17, and pushouts of fair C∞-rings need not be fair. Our next result is referred to in the last part of Dubuc [23, Th. 13]. Proposition 2.26. C∞Ringsfais not closed under pushouts in C∞Rings. Nonetheless, pushouts and all finite colimits exist in C∞Ringsfa,although they may not coincide with pushouts and finite colimits in C∞Rings. Proof.Example 2.25 shows that C∞Ringsfais not closed under pushouts in C∞Rings. To construct finite colimits in C∞Ringsfa, we first take the colimit inC∞Ringsfg, which exists by Propositions 2.5 and 2.24, and then apply the reflection functor Rfa fg. By the universal properties of colimits and reflection functors, the result is a colimit in C∞Ringsfa. 2.6 Flat ideals The following class of ideals in C∞(Rn) is defined by Moerdijk and Reyes [54, p. 47, p. 49] (see also Dubuc [22, §1.7(a)]), who call them flat ideals : Definition 2.27. LetXbe a closed subset of Rn. Define m∞ Xto be the ideal of all functions g∈C∞(Rn) such that ∂kg|X≡0 for allk/greaterorequalslant0, that is,gand all its derivatives vanish at each x∈X. If the interior X◦ofXinRnis dense inX, that is (X◦) =X, then∂kg|X≡0 for allk/greaterorequalslant0 if and only if g|X≡0. In this caseC∞(Rn)/m∞ X∼=C∞(X) :=/braceleftbig f|X:f∈C∞(Rn)/bracerightbig . Flat ideals are always fair. Here is an example from [54, Th. I.1.3]. Example 2.28. TakeXtobethepoint{0}. Iff,f′∈C∞(Rn)thenf−f′liesin m∞ {0}if and only if f,f′have the same Taylor series at 0. Thus C∞(Rn)/m∞ {0}is theC∞-ring of Taylor series at 0 of f∈C∞(Rn). Since any formal power series inx1,...,xnis the Taylorseries of some f∈C∞(Rn), we haveC∞(Rn)/m∞ {0}∼= R[[x1,...,xn]]. Thus the R-algebra of formal power series R[[x1,...,xn]] can be made into a C∞-ring. 16The following nontrivial result is proved by Reyes and van Quˆ e [60, Th . 1], generalizing an unpublished result of A.P. Calder´ on in the case X=Y={0}. It can also be found in Moerdijk and Reyes [54, Cor. I.4.12]. Proposition 2.29. LetX⊆RmandY⊆Rnbe closed. Then as ideals in C∞(Rm+n)we have (m∞ X,m∞ Y) =m∞ X×Y. Moerdijk and Reyes [54, Cor. I.4.19] prove: Proposition 2.30. LetX⊆Rnbe closed with X/\e}atio\slash=∅,Rn. Then the ideal m∞ X inC∞(Rn)is not countably generated. We can use these to study C∞-rings of manifolds with corners. Example 2.31. Let 00, we can embedXasaclosedsubsetinan n-manifoldX′withoutboundary,suchthatthe inclusionX ֒→X′is locally modelled on the inclusion of Rn k= [0,∞)k×Rn−kin (−ǫ,∞)k×Rn−kfork/lessorequalslantn. Choose a closed embedding i:X′֒→RNforN≫0 as above, giving 0 →I′→C∞(RN)i∗ −→C∞(X′)→0 withI′generated by f1,...,fk∈C∞(RN). Leti(X′)⊂T⊂RNbe an open tubular neighbourhood ofi(X′) inRN, with projection π:T→i(X′). SetU=π−1(i(X◦))⊂T⊂RN, whereX◦is the interior of X. ThenUis open in RNwithi(X◦) =U∩i(X′), and the closure ¯UofUinRNhasi(X) =¯U∩i(X′). LetIbe the ideal ( f1,...,fk,m∞¯U) inC∞(RN). ThenIis fair, as (f1,...,fk) andm∞¯Uare fair. Since Uis open in RNand dense in ¯U, as in Definition 2.27 we have g∈m∞ ¯Uif and only if g|¯U≡0. Therefore the isomorphism (i∗)∗:C∞(RN)/I′→C∞(X′) identifies the ideal I/I′inC∞(X′) with the ideal off∈C∞(X′) such that f|X≡0, sinceX=i−1(¯U). Hence C∞(RN)/I∼=C∞(X′)//braceleftbig f∈C∞(X′) :f|X≡0/bracerightbig∼=/braceleftbig f|X:f∈C∞(X′)/bracerightbig∼=C∞(X). AsIis a fair ideal, this implies that C∞(X) is a fairC∞-ring. If∂X/\e}atio\slash=∅then using Proposition 2.30 we can show Iis not countably generated, so C∞(X) is not finitely presented by Proposition 2.17. Next we consider the transformation X/ma√sto→C∞(X) as a functor. Definition 3.2. WriteC∞Ringsop, (C∞Ringsfp)op, (C∞Ringsfa)opfor the opposite categories of C∞Rings,C∞Ringsfp,C∞Ringsfa(i.e. directions of morphisms are reversed). Define functors FC∞Rings Man:Man−→(C∞Ringsfp)op⊂C∞Ringsop, FC∞Rings Manb:Manb−→(C∞Ringsfa)op⊂C∞Ringsop, FC∞Rings Manc:Manc−→(C∞Ringsfa)op⊂C∞Ringsop asfollows. On objectsthe functors FC∞Rings Man∗mapX/ma√sto→C∞(X), whereC∞(X) is aC∞-ring as in Example 2.2. On morphisms, if f:X→Yis a smooth map of manifolds then f∗:C∞(Y)→C∞(X) mapping c/ma√sto→c◦fis a morphism 18ofC∞-rings, so that f∗:C∞(Y)→C∞(X) is a morphism in C∞Rings, andf∗:C∞(X)→C∞(Y) a morphism in C∞Ringsop, andFC∞Rings Man∗maps f/ma√sto→f∗. ClearlyFC∞Rings Man,FC∞Rings Manb,FC∞Rings Mancare functors. Iff:X→Yisonlyweakly smooth thenf∗:C∞(Y)→C∞(X)inDefinition 3.2 is still a morphism of C∞-rings. From [54, Prop. I.1.5] we deduce: Proposition 3.3. LetX,Ybe manifolds with corners. Then the map f/ma√sto→f∗ from weakly smooth maps f:X→Yto morphisms of C∞-ringsφ:C∞(Y)→ C∞(X)is a1-1correspondence. In the category of manifolds Man, the morphisms are weakly smooth maps. SoFC∞Rings Man is both injective on morphisms (faithful), and surjective on mor- phisms (full), as in Moerdijk and Reyes [54, Th. I.2.8]. But in Manb,Manc the morphisms are smooth maps, a proper subset of weakly smooth maps, so the functors are injective but not surjective on morphisms. That is: Corollary 3.4. The functor FC∞Rings Man:Man→(C∞Ringsfp)opis full and faithful. However, the functors FC∞Rings Manb:Manb→(C∞Ringsfa)opand FC∞Rings Manc:Manc→(C∞Ringsfa)opare faithful, but not full. Of course, if we defined Manb,Mancto have morphisms weakly smooth maps, then FC∞Rings Manb,FC∞Rings Mancwould be full and faithful. LetX,Y,Zbe manifolds and f:X→Z,g:Y→Zbe smooth maps. If X,Y,Zare without boundary then f,gare called transverse if whenever x∈X andy∈Ywithf(x) =g(y) =z∈Zwe haveTzZ= df(TxX)+dg(TyY). If f,gare transverse then a fibre product X×ZYexists in Man. For manifolds with boundary, or with corners, the situation is more c ompli- cated, as explained in [35, §6], [40,§4.3]. In the definition of smoothf:X→Y we impose extra conditions over ∂jX,∂kY, and in the definition of transverse f,gwe impose extra conditions over ∂jX,∂kY,∂lZ. With these more restrictive definitions of smooth and transverse maps, transverse fibre pro ducts exist in Mancby [35, Th. 6.3] (see also [40, Th. 4.27]). The na¨ ıve definition of tran sver- sality is not a sufficient condition for fibre products to exist. Note to o that a fibre product of manifolds with boundary may be a manifold with corne rs, so fibre products work best in ManorMancrather than Manb. Our next theorem is given in [23, Th. 16] and [54, Prop. I.2.6] for manif olds without boundary, and the special case of products Man×Manb→Manb follows from Reyes [59, Th. 2.5], see also Kock [44, §III.9]. It can be proved by combining the usual proof in the without boundary case, the pro of of [35, Th. 6.3], and Proposition 2.29. Theorem 3.5. The functors FC∞Rings Man,FC∞Rings Mancpreserve transverse fibre products in Man,Manc,in the sense of [35,§6]. That is, if the following is a Cartesian square of manifolds with g,htransverse Wf/d47/d47 e/d15/d15Y h/d15/d15 Xg/d47/d47Z,(3.1) 19so thatW=X×g,Z,hY,then we have a pushout square of C∞-rings C∞(Z) h∗/d47/d47 g∗/d15/d15C∞(Y) f∗/d15/d15 C∞(X)e∗/d47/d47C∞(W),(3.2) so thatC∞(W) =C∞(X)∐g∗,C∞(Z),h∗C∞(Y). 4C∞-ringed spaces and C∞-schemes Inalgebraicgeometry,if Aisanaffineschemeand Rtheringofregularfunctions onA, then we can recover Aas the spectrum of the ring R,A∼=SpecR. One of the ideas of synthetic differential geometry, as in [54, §I], is to regard a manifold Xas the ‘spectrum’ of the C∞-ringC∞(X) in Example 2.2. So we can try to develop analogues of the tools of scheme theory for smooth manifo lds, replacing rings byC∞-rings throughout. This was done by Dubuc [22,23]. The analogues of the algebraic geometry notions [31, §II.2] of ringed spaces, locally ringed spaces, and schemes, are called C∞-ringed spaces, local C∞-ringed spaces and C∞-schemes. The material of §4.6–§4.9 is new. 4.1 Some basic topology Later we will use several properties of topological spaces, e.g. se cond countable, metrizable, Lindel¨ of, ..., so we now recall their definitions and som e relation- ships between them. Let Xbe a topological space, with topology T. Then: •AbasisforTis a familyB⊆Tsuch that every open set in Xis a union of sets inB. We callXsecond countable ifThas a countable basis. •An open cover{Ui:i∈I}ofXislocally finite if everyx∈Xhas an open neighbourhood WwithW∩Ui/\e}atio\slash=∅for only finitely many i∈I. An open cover{Vj:j∈J}ofXis arefinement of another open cover {Ui:i∈I}if for allj∈Jthere exists i∈IwithVj⊆Ui⊆X. We callXparacompact if every open cover {Ui:i∈I}ofXadmits a locally finite refinement {Vj:j∈J}. •We callXHausdorff if for allx,y∈Xwithx/\e}atio\slash=ythere exist open U,V⊆Xwithx∈U,y∈VandU∩V=∅. •We callXmetrizable if there exists a metric on Xinducing topology T. •We callXregularif for every closed subset C⊆Xand eachx∈X\C there exist disjoint open sets U,V⊆XwithC⊆Uandx∈V. •We callXcompletely regular if for every closed C⊆Xandx∈X\C there exists a continuous f:X→[0,1] withf|C= 0 andf(x) = 1. •We callXseparable if it has a countable dense subset S⊆X. 20•We callXlocally compact if for allx∈Xthere exist x∈U⊆C⊆X withUopen andCcompact. •We callXLindel¨ of if every open cover of Xhas a countable subcover. By well known results in topology, including Urysohn’s metrization the orem, the following are equivalent: (i)Xis Hausdorff, second countable and regular. (ii)Xis second countable and metrizable. (iii)Xis separable and metrizable. Here are some useful implications: •XHausdorff and locally compact imply Xis regular. •Xmetrizable implies Xis Hausdorff, paracompact, and regular. •Xsecond countable implies Xis Lindel¨ of. •XLindel¨ of and regular imply Xis paracompact. 4.2 Sheaves on topological spaces Sheaves are a fundamental concept in algebraic geometry. They a re necessary even to define schemes, since a scheme is a topological space Xequipped with a sheaf of ringsOX. In this book, sheaves of C∞-rings, and sheaves of modules over a sheaf of C∞-rings, play a fundamental rˆ ole. We now summarize some basics of sheaf theory, following Hartshorn e [31, §II.1]. A more detailed reference is Godement [28]. We concentrate on sheaves of abelian groups; to define sheaves of C∞-rings, etc., one replaces abelian groups with C∞-rings, etc., throughout. This is justified since limits in all these categories (including abelian groups) are computed at the level of u nderlying sets, because they are all algebras for algebraic theories. Definition 4.1. LetXbe a topological space. A presheaf of abelian groups E onXconsists of the data of an abelian group E(U) for every open set U⊆X, and a morphism of abelian groups ρUV:E(U)→E(V) called the restriction mapfor every inclusion V⊆U⊆Xof open sets, satisfying the conditions that (i)E(∅) = 0; (ii)ρUU= idE(U):E(U)→E(U) for all open U⊆X; and (iii)ρUW=ρVW◦ρUV:E(U)→E(W) for all open W⊆V⊆U⊆X. That is, a presheaf is a functor E:Open(X)op→AbGp, whereOpen(X) is the category of open subsets of Xwith morphisms inclusions, and AbGpis the category of abelian groups. A presheaf of abelian groups EonXis called a sheafif it also satisfies (iv) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, ands∈E(U) has ρUVi(s) = 0 inE(Vi) for alli∈I, thens= 0 inE(U); and 21(v) IfU⊆Xis open,{Vi:i∈I}is an open cover of U, and we are given elementssi∈E(Vi) for alli∈Isuch thatρVi(Vi∩Vj)(si) =ρVj(Vi∩Vj)(sj) inE(Vi∩Vj) for alli,j∈I, then there exists s∈E(U) withρUVi(s) =si for alli∈I. Thissis unique by (iv). SupposeE,Fare presheavesor sheavesof abelian groups on X. Amorphism φ:E→Fconsists of a morphism of abelian groups φ(U) :E(U)→F(U) for all openU⊆X, suchthatthefollowingdiagramcommutesforallopen V⊆U⊆X E(U) φ(U)/d47/d47 ρUV/d15/d15F(U) ρ′ UV/d15/d15 E(V)φ(V)/d47/d47F(V), whereρUVis the restriction map for E, andρ′ UVthe restriction map for F. Definition 4.2. LetEbe a presheaf of abelian groups on X. For eachx∈X, thestalkExis the direct limit of the groups E(U) for allx∈U⊆X, via the restriction maps ρUV. It is an abelian group. A morphism φ:E→Finduces morphisms φx:Ex→Fxfor allx∈X. IfE,Fare sheaves then φis an isomorphism if and only if φxis an isomorphism for all x∈X. Sheaves of abelian groups on Xform an abelian category Sh(X). Thus we have (category-theoretic) notions of when a morphism φ:E→Fin Sh(X) is injective orsurjective (epimorphic ), and when a sequence E→F→G in Sh(X) isexact. It turns out that φ:E→Fis injective if and only if φ(U) :E(U)→ F(U) is injective for all open U⊆X. Howeverφ:E→Fsurjective does not imply that φ(U) :E(U)→F(U) is surjective for all open U⊆X. Instead,φis surjective if and only if φx:Ex→Fxis surjective for all x∈X. Definition 4.3. LetEbe a presheaf of abelian groups on X. Asheafification ofEis a sheaf of abelian groups ˆEonXand a morphism π:E→ˆE, such that wheneverFis a sheaf of abelian groups on Xandφ:E→Fis a morphism, there is a unique morphism ˆφ:ˆE→Fwithφ=ˆφ◦π. As in [31, Prop. II.1.2], a sheafification always exists, and is unique up to canonical isomorph ism; one can be constructed explicitly using the stalks ExofE. Next we discuss pushforwards andpullbacks of sheaves by continuous maps. Definition 4.4. Letf:X→Ybe a continuous map of topological spaces, and Ea sheaf of abelian groups on X. Define the pushforward (direct image ) sheaf f∗(E) onYby/parenleftbig f∗(E)/parenrightbig (U) =E/parenleftbig f−1(U)/parenrightbig for all open U⊆V, with restriction mapsρ′ UV=ρf−1(U)f−1(V):/parenleftbig f∗(E)/parenrightbig (U)→/parenleftbig f∗(E)/parenrightbig (V) for all open V⊆U⊆Y. Thenf∗(E) is a sheaf of abelian groups on Y. Ifφ:E→Fis a morphism in Sh( X) we define f∗(φ) :f∗(E)→f∗(F) by/parenleftbig f∗(φ)/parenrightbig (u) =φ/parenleftbig f−1(U)/parenrightbig for all open U⊆Y. Thenf∗(φ) is a morphism in Sh(Y), andf∗is a functor Sh( X)→Sh(Y). It is a left exact functor between abelian categories, but in general is not exact. For continuous map sf:X→Y, g:Y→Zwe have (g◦f)∗=g∗◦f∗. 22Definition 4.5. Letf:X→Ybe a continuous map of topological spaces, andEa sheaf of abelian groups on Y. Define a presheaf Pf−1(E) onXby/parenleftbig Pf−1(E)/parenrightbig (U) = limA⊇f(U)E(A) for open A⊆X, where the direct limit is taken over all open A⊆Ycontaining f(U), using the restriction maps ρAB inE. For open V⊆U⊆X, defineρ′ UV:/parenleftbig Pf−1(E)/parenrightbig (U)→/parenleftbig Pf−1(E)/parenrightbig (V) as the direct limit of the morphisms ρABinEforB⊆A⊆Ywithf(U)⊆A andf(V)⊆B. Then we define the pullback (inverse image )f−1(E) to be the sheafification of the presheaf Pf−1(E). Pullbacksf−1(E) are only unique up to canonical isomorphism, rather than unique. By convention we choose once and for all a pullback f−1(E) for all X,Y,f,E, using the Axiom of Choice if necessary. If φ:E→Fis a morphism in Sh(Y), one can define a pullback morphism f−1(φ) :f−1(E)→f−1(F). Thenf−1: Sh(Y)→Sh(X) is an exact functor between abelian categories. We compare pushforwards and pullbacks: Remark 4.6. (a) There are two kinds of pullback, with slightly different no- tation. The first kind, written f−1(E) as in Definition 4.5, is used for sheaves of abelian groups or C∞-rings. The second kind, written f∗(E) orf∗(E) and discussed in§5.3 and§8.3, is used for sheaves of OY-modulesE. (b)The definition of pushforward sheaves f∗(E) is wholly elementary. In con- trast, the definition of pullbacks f−1(E) is complex, involving a direct limit followed by a sheafification, and includes arbitrary choices. Pushforwards f∗are strictly functorial in the continuous map f:X→Y, thatis, forcontinuous f:X→Y,g:Y→Zwehave(g◦f)∗=g∗◦f∗: Sh(X)→ Sh(Z). However, pullbacks f−1are only weakly functorial in f: ifE∈Sh(Z) then we need not have ( g◦f)−1(E) =f−1(g−1(E)). This is because pullbacks are only natural up to canonical isomorphism, and we make an arbitr ary choice for each pullback. So although f−1(g−1(E)) is a possible pullback for Ebyg◦f, it may not be the one we chose. Thus, thereisacanonicalisomorphism( g◦f)−1(E)∼=f−1(g−1(E)), whichwe will write as If,g(E) : (g◦f)−1(E)→f−1(g−1(E)). TheIf,g(E) for allE∈Sh(Z) comprise a natural isomorphism of functors If,g: (g◦f)−1⇒f−1◦g−1. Sim- ilarly, forE ∈Sh(X) we may not have id−1 X(E) =E, but instead there are canonical isomorphisms δX(E) : id−1 X(E)→E, which make up a natural iso- morphismδX: id−1 X⇒idSh(X). Many authors ignore the natural isomorphisms If,g,δXentirely. (c)Letf:X→Ybe a continuous map of topological spaces. Then we have functorsf∗: Sh(X)→Sh(Y), andf−1: Sh(Y)→Sh(X). Asin[31,Ex.II.1.18], f∗is right adjoint to f−1. That is, there is a natural bijection HomX/parenleftbig f−1(E),F/parenrightbig∼=HomY/parenleftbig E,f∗(F)/parenrightbig (4.1) for allE∈Sh(Y) andF∈Sh(X), with functorial properties. We define finesheaves, as in Godement [28, §II.3.7] or Voisin [69, Def. 4.35]. They will be important in §4.7 and§5.3. 23Definition 4.7. LetXbe a topological space (usually paracompact), and Ea sheaf of abelian groups on X, or more generally a sheaf of rings, or C∞-rings, orOX-modules, or any other objects which are also abelian groups. We ca llE fineif for any open cover {Ui:i∈I}ofX, a subordinate locally finite partition of unity{ζi:i∈I}exists in the sheaf Hom(E,E). Hereζi:E→Eis a morphism of sheaves of abelian groups (or rings, C∞- rings, ...) for each i∈I. For{ζi:i∈I}to besubordinate to{Ui:i∈I} means that ζiis supported in Uifor eachi∈I, that is, there exists open Vi⊆X withζi|Vi= 0 andUi∪Vi=X. For{ζi:i∈I}to belocally finite means that eachx∈Xhas an open neighbourhood Wwithζi|W/\e}atio\slash= 0 for only finitely many i∈I. For{ζi:i∈I}to be apartition of unity means that/summationtext i∈Iζi= idE, where the sum makes sense as {ζi:i∈I}is locally finite. IfE=OXis a sheaf of commutative rings or C∞-rings, then writing ηi= ζi(1) inOX(X), we see that ζi=ηi·is multiplication by ηi. So we can regard the partition of unity as living in OX(X) rather thanHom(OX,OX). 4.3C∞-ringed spaces and local C∞-ringed spaces Definition 4.8. AC∞-ringed space X= (X,OX) is a topological space X with a sheafOXofC∞-rings onX. That is, for each open set U⊆Xwe are given aC∞ringOX(U), and for each inclusion of open sets V⊆U⊆Xwe are given a morphism of C∞-ringsρUV:OX(U)→OX(V), called the restriction maps, and all this data satisfies the sheaf axioms in Definition 4.1. Equivalently,OXis a presheaf of C∞-rings onX, that is, a functor OX:Open(X)op−→C∞Rings, whose underlying presheaf of abelian groups, or of sets, is a sheaf . The sheaf axioms Definition 4.1(iv),(v) do not use the C∞-ring structure. Amorphismf= (f,f♯) : (X,OX)→(Y,OY) ofC∞ringed spaces is a continuous map f:X→Yand a morphism f♯:f−1(OY)→OXof sheaves of C∞-rings onX, forf−1(OY) as in Definition 4.5. Since f∗is right adjoint to f−1, as in (4.1) there is a natural bijection HomX/parenleftbig f−1(OY),OX/parenrightbig∼=HomY/parenleftbig OY,f∗(OX)/parenrightbig . (4.2) Writef♯:OY→f∗(OX) for the morphism of sheaves of C∞-rings onYcorre- sponding to f♯under (4.2), so that f♯:f−1(OY)−→OX/squiggleleftrightf♯:OY−→f∗(OX). (4.3) Iff:X→Yandg:Y→ZareC∞-scheme morphisms, the composition is g◦f=/parenleftbig g◦f,(g◦f)♯/parenrightbig =/parenleftbig g◦f,f♯◦f−1(g♯)◦If,g(OZ)/parenrightbig , whereIf,g(OZ) : (g◦f)−1(OZ)→f−1(g−1(OZ)) is the canonical isomorphism from Remark 4.6(b). In terms of f♯:OY→f∗(OX), composition is (g◦f)♯=g∗(f♯)◦g♯:OZ−→(g◦f)∗(OX) =g∗◦f∗(OX). 24AlocalC∞-ringed space X= (X,OX) is aC∞-ringed space for which the stalksOX,xofOXatxare localC∞-rings for all x∈X. As in Definition 2.10, since morphisms of local C∞-rings are automatically local morphisms, morphisms of local C∞-ringed spaces ( X,OX),(Y,OY) are just morphisms of C∞-ringedspaces,without anyadditionallocalitycondition. Moerdijk, vanQuˆ e and Reyes [52,§3] call our local C∞-ringed spaces Archimedean C∞-spaces. WriteC∞RSfor the category of C∞-ringed spaces, and LC∞RSfor the full subcategory of local C∞-ringed spaces. For brevity, we will use the notation that underlined upper case lett ers X,Y,Z,...representC∞-ringed spaces ( X,OX),(Y,OY),(Z,OZ),...,and un- derlined lower case letters f,g,...represent morphisms of C∞-ringed spaces (f,f♯),(g,g♯),....When we write ‘ x∈X’ we mean that X= (X,OX) and x∈X. When we write ‘ Uis open in X’ we mean that U= (U,OU) and X= (X,OX) withU⊆Xan open set andOU=OX|U. Remark 4.9. As above, there are two equivalent ways to write morphisms ofC∞-ringed spaces ( X,OX)→(Y,OY), either using pullbacks as ( f,f♯) for f♯:f−1(OY)→OX, or using pushforwards as ( f,f♯) forf♯:OY→f∗(OX). Each definition has advantages and disadvantages. We choose to r egardf♯: f−1(OY)→OXas the primary object, and so define morphisms of C∞-ringed spaces as (f,f♯) rather than ( f,f♯), although we will use f♯in a few places. We can always switch between the two points of view using (4.3). Example 4.10. LetXbe a manifold, which may have boundary or corners. Define aC∞-ringed space X= (X,OX) to have topological space Xand OX(U) =C∞(U) for each open subset U⊆X, whereC∞(U) is theC∞- ring of smooth maps c:U→R, and ifV⊆U⊆Xare open we define ρUV:C∞(U)→C∞(V) byρUV:c/ma√sto→c|V. It is easyto verify that OXis a sheaf of C∞-ringsonX(not just a presheaf), soX= (X,OX) is aC∞-ringed space. For each x∈X, the stalkOX,xis the localC∞-ring of germs [( c,U)] of smooth functions c:X→Ratx∈X, as in Example 2.15, with unique maximal ideal mX,x=/braceleftbig [(c,U)]∈OX,x:c(x) = 0/bracerightbig andOX,x/mX,x∼=R. HenceXis a localC∞-ringed space. LetX,Ybe manifolds and f:X→Ya weakly smooth map. Define (X,OX),(Y,OY) as above. For all open U⊆Ydefinef♯(U) :OY(U) = C∞(U)→OX(f−1(U)) =C∞(f−1(U)) byf♯(U) :c/ma√sto→c◦ffor allc∈C∞(U). Thenf♯(U) is a morphism of C∞-rings, and f♯:OY→f∗(OX) is a morphism of sheaves of C∞-rings onY. Letf♯:f−1(OY)→OXcorrespond to f♯un- der (4.3). Then f= (f,f♯) : (X,OX)→(Y,OY) is a morphism of (local) C∞-ringed spaces. As the category Topof topological spaces has all finite limits, and the con- structionof C∞RSinvolvesTopinacovariantwayandthecategory C∞Rings in a contravariant way, using Proposition 2.5 one may prove: Proposition 4.11. All finite limits exist in the category C∞RS. Dubuc [23, Prop. 7] proves: 25Proposition 4.12. The full subcategory LC∞RSof localC∞-ringed spaces in C∞RSis closed under finite limits in C∞RS. 4.4 The spectrum functor We now define a spectrum functor Spec :C∞Ringsop→LC∞RS. It is equivalent to those constructed by Dubuc [22,23] and Moerdijk, v an Quˆ e and Reyes [52,§3], but our presentation is closer to that of Hartshorne [31, p. 70]. Definition 4.13. LetCbe aC∞-ring, and use the notation of Definition 2.13. WriteXCfor the set of all R-pointsxofC. LetTCbe the topology on XC generated by the basis of open sets Uc=/braceleftbig x∈XC:x(c)/\e}atio\slash= 0/bracerightbig for allc∈C. For eachc∈Cdefinec∗:XC→Rto mapc∗:x/ma√sto→x(c). Example 4.14. Suppose Cis a finitely generated C∞-ring, with exact sequence 0→I ֒→C∞(Rn)φ−→C→0. Define a map φ∗:XC→Rnbyφ∗:x/ma√sto→/parenleftbig x◦φ(x1),...,x◦φ(xn)/parenrightbig , wherex1,...,xnare the generators of C∞(Rn). Then φ∗gives a homeomorphism φ∗:XC∼=−→Xφ C=/braceleftbig (x1,...,xn)∈Rn:f(x1,...,xn) = 0 for all f∈I/bracerightbig ,(4.4) where the right hand side is a closed subset of Rn. So the topological spaces (XC,TC) for finitely generated Care homeomorphic to closed subsets of Rn. Recall that a topological space Xisregularif whenever S⊆Xis closed and x∈X\Sthen there exist open U,V⊆Xwithx∈U,S⊆VandU∩V=∅. Lemma 4.15. In Definition 4.13,the topologyTCis also generated by the basis of open sets c−1 ∗(V)for allc∈Cand openV⊆R. That is,TCis the weakest topology on XCsuch thatc∗:XC→Ris continuous for all c∈C. Also (XC,TC)is a Hausdorff, regular topological space. Proof.Supposec∈CandV⊆Ris open. Then there exists smooth f:R→R withV={x∈R:f(x)/\e}atio\slash= 0}. Setc′= Φf(c), using the C∞-ring operation Φf:C→C. Thenc′ ∗=f◦c∗asc:C→Ris aC∞-ring morphism, so Uc′= (c′ ∗)−1(R\{0}) = (f◦c∗)−1(R\{0}) =c−1 ∗[f−1(0)] =c−1 ∗(V). Soc−1 ∗(V) is of the form Uc′. Conversely Uc=c−1 ∗(V) forV=R\{0}⊆R. So the two given bases for TCare the same, proving the first part. Letx,ybe distinct points of XC. Then there exists c∈Cwithx(c)/\e}atio\slash=y(c), asx/\e}atio\slash=y. Setǫ=1 2|x(c)−y(c)|>0 andU=c−1 ∗/parenleftbig (x(c)−ǫ,x(c) +ǫ)/parenrightbig , V=c−1 ∗/parenleftbig (y(c)−ǫ,y(c) +ǫ)/parenrightbig . ThenU,V⊆XCare disjoint open sets with x∈U,y∈V, so (XC,TC) is Hausdorff. SupposeS⊆XCis closed, and x∈X\S. Then there exists c∈Cwithx∈ Uc⊆XC\S, sinceXC\Sis open inXCand theUcare a basis forTC. Therefore c∗(x)/\e}atio\slash= 0 andc∗|S= 0. Setǫ=1 2|c∗(x)|>0,U=c−1 ∗/parenleftbig (c∗(x)−ǫ,c∗(x) +ǫ)/parenrightbig andV=c−1 ∗/parenleftbig (−ǫ,ǫ)/parenrightbig . ThenU,V⊆XCare disjoint open sets with x∈U, S⊆V, so (XC,TC) is regular. 26Definition 4.16. LetCbe aC∞-ring, and XCthe topological space from Definition4.13. Foreachopen U⊆XC, defineOXC(U) tobe thesetoffunctions s:U→/coproducttext x∈UCxwiths(x)∈Cxfor allx∈U, and such that Umay be covered by open sets W⊆U⊆XCfor which there exist c∈Cwiths(x) =πx(c)∈Cx for allx∈W. Define operations Φ fonOXC(U) pointwise in x∈Uusing the operations Φ fonCx. This makesOXC(U) into aC∞-ring. IfV⊆U⊆XCare open, the restriction map ρUV:OXC(U)→OXC(V) mappingρUV:s/ma√sto→s|Vis a morphism of C∞-rings. ClearlyOXCis a sheaf of C∞-rings onXC. Lemma 4.18 shows that the stalk OXC,xatx∈XCisCx, which is a local C∞-ring. Hence ( XC,OXC) is a local C∞-ringed space, which we call the spectrum ofC, and write as Spec C. Now letφ:C→Dbe a morphism of C∞-rings. Define fφ:XD→ XCbyfφ(x) =x◦φ. Thenfφis continuous. For U⊆XCopen define (fφ)♯(U) :OXC(U)→OXD(f−1 φ(U)) by (fφ)♯(U)s:x/ma√sto→φx(s(fφ(x))), where φx:Cfφ(x)→Dxis the induced morphism of local C∞-rings. Then ( fφ)♯: OXC→(fφ)∗(OXD) is a morphism of sheaves of C∞-rings onXC. Letf♯ φ: f−1 φ(OXC)→OXDbe the corresponding morphism of sheaves of C∞-rings on XDunder (4.3). Then fφ= (fφ,f♯ φ) : (XD,OXD)→(XC,OXC) is a morphism of localC∞-ringed spaces. Define Spec φ: SpecD→SpecCby Specφ=fφ. Then Spec is a functor C∞Ringsop→LC∞RS, thespectrum functor . Example 4.17. LetXbe a manifold. Then it followsfrom Theorem 4.41below that the local C∞-ringed space Xconstructed in Example 4.10 is naturally isomorphic to Spec C∞(X). Lemma 4.18. In Definition 4.16,the stalkOXC,xofOXCatx∈XCis nat- urally isomorphic to Cx. Proof.Elements ofOXC,xare∼-equivalence classes [ U,s] of pairs (U,s), where Uis an open neighbourhood of xinXCands∈OXC(U), and (U,s)∼(U′,s′) if there exists open x∈V⊆U∩U′withs|V=s′|V. Define aC∞-ring morphism Π :OXC,x→Cxby Π : [U,s]/ma√sto→s(x). Supposecx∈Cx. Thencx=πx(c) for some c∈Cby Proposition 2.14. The maps:XC→/coproducttext x′∈XCCx′mappings:x′/ma√sto→πx′(c) lies inOXC(XC), and Π : [XC,s]/ma√sto→πx(c) =cx. Hence Π :OXC,x→Cxis surjective. Suppose [U,s]∈OXC,xwith Π([U,s]) = 0∈Cx. Ass∈OXC(U), there exist openx∈V⊆Uandc∈Cwiths(x′) =πx′(c)∈Cx′for allx′∈V. Then πx(c) =s(x) = Π([U,s]) = 0, soclies in the ideal Iin (2.2) by Proposition 2.14. Thus there exists d∈Cwithx(d)/\e}atio\slash= 0 inRandcd= 0 inC. Set W={x′∈V:x′(d)/\e}atio\slash= 0}, so thatWis an open neighbourhood of xinU. If x′∈Wthenx′(d)/\e}atio\slash= 0, soπx′(d) is invertible in Cx′. Thus s(x′) =πx′(c) =πx′(c)πx′(d)πx′(d)−1=πx′(cd)πx′(d)−1=πx′(0)πx′(d)−1= 0. Hence [U,s] = [W,s|W] = [W,0] = 0 inOXC,x, so Π :OXC,x→Cxis injective. Thus Π :OXC,x→Cxis an isomorphism. 27Definition 4.19. Theglobal sections functor Γ :LC∞RS→C∞Ringsop acts on objects ( X,OX) by Γ : (X,OX)/ma√sto→OX(X) and on morphisms ( f,f♯) : (X,OX)→(Y,OY) by Γ : (f,f♯)/ma√sto→f♯(Y), forf♯:OY→f∗(OX) as in (4.3). Then Γ◦Spec is a functor C∞Ringsop→C∞Ringsop, or equivalently a functorC∞Rings→C∞Rings. For eachC∞-ringCandc∈C, define Ψ C(c) : XC→/coproducttext x∈XCCxby ΨC(c) :x/ma√sto→πx(c)∈Cx. Then Ψ C(c)∈OXC(XC) = Γ◦SpecCby Definition 4.16, soΨ C:C→Γ◦SpecCis amap. Since πx:C→Cx is aC∞-ring morphism and the C∞-ring operations on OXC(XC) are defined pointwise in the Cx, this Ψ Cis aC∞-ring morphism. It is functorial in C, so that the Ψ Cfor allCdefine a natural transformation Ψ : id C∞Rings⇒Γ◦Spec of functors id C∞Rings,Γ◦Spec :C∞Rings→C∞Rings. Theorem 4.20. The functor Spec :C∞Ringsop→LC∞RSisright adjoint toΓ :LC∞RS→C∞Ringsop. That is, for all C∈C∞RingsandX∈ LC∞RSthere are inverse bijections HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC), RC,X/d111/d111 (4.5) which are functorial in the sense that if λ:C→Dis a morphism in C∞Rings ande:X→Ya morphism in LC∞RSthen the following commutes: HomC∞Rings(D,Γ(Y))LD,Y/d47/d47 φ/mapsto→Γ(e)◦φ◦λ/d15/d15HomLC∞RS(Y,SpecD) RD,Y/d111/d111 f/mapsto→Specλ◦f◦e/d15/d15 HomC∞Rings(C,Γ(X))LC,X/d47/d47HomLC∞RS(X,SpecC). RC,X/d111/d111(4.6) WhenX= SpecCwe have ΨC=RC,X(idX),so thatΨCis the unit of the adjunction between ΓandSpec. Proof.LetC∈C∞RingsandX∈LC∞RS. WriteY= (Y,OY) = Spec C. DefineRC,Xin (4.5) by, for each morphism f:X→YinLC∞RS, taking RC,X(f) :C→Γ(X) to be the composition CΨC/d47/d47Γ◦SpecC= Γ(Y)Γ(f)/d47/d47Γ(X). (4.7) For the last part, if X= SpecCthen Ψ C=RC,X(idX) as Γ(idX) = idΓ(X). Letφ:C→Γ(X) be a morphism in C∞Rings. We will construct a morphismg= (g,g♯) :X→YinLC∞RS, and setLC,X(φ) =g. For any x∈Xwe have an R-algebra morphism x∗: Γ(X)→Rby composing the stalk mapσx: Γ(X)→OX,xwith the unique morphism π:OX,x→R, asOX,xis a localC∞-ring. Then x∗◦φ:C→RisanR-algebramorphism, andhenceapoint ofY. Defineg:X→Ybyg(x) =x∗◦φ. Ifc∈CthenUc={y∈Y:y(c)/\e}atio\slash= 0} is open inY, andg−1(Uc) ={x∈X:x∗(φ(c))/\e}atio\slash= 0}is open inX, asx/ma√sto→x∗(d) is a continuous map X→Rfor anyd∈Γ(X). Since the Ucforc∈Care a basis for the topology of Yby Definition 4.13, g:X→Yis continuous. 28Letx∈Xwithg(x) =y∈Y. Consider the diagram of C∞-rings C πy/d15/d15φ/d47/d47Γ(X) σx/d15/d15 Cy∼=OY,yφx/d47/d47OX,x.(4.8) HereCy∼=OY,yby Lemma 4.18. If c∈Cwithy(c)/\e}atio\slash= 0 thenσx◦φ(c)∈OX,x withπ[σx◦φ(c)]/\e}atio\slash= 0, soσx◦φ(c) is invertible inOX,xasOX,xis a localC∞- ring. Thus by the universal property of πy:C→Cythere is a unique morphism φx:OY,y→OX,xmaking (4.8) commute. For each open V⊆YwithU=g−1(V)⊆X, defineg♯(V) :OY(V)→ g∗(OX)(V) =OX(U) byg♯(V)s:x/ma√sto→φx(s(g(x))) fors∈OY(V) andx∈U⊆ X, so thatg(x)∈V, ands(g(x))∈OY,g(x), andφx(s(g(x)))∈OX,x. Here as OXisasheafwemayidentify elementsof OX(U)withmaps t:U→/coproducttext x∈UOX,x witht(x)∈OX,xforx∈U, such that tsatisfies certain local conditions in U. Ifs∈OY(V) andx∈U⊆Xwithg(x) =y∈V⊆Y, then by Definition 4.16 there is an open neighbourhood WyofyinVandc∈Cwiths(y′) =πy′(c)∈ Cy′∼=OY,y′for ally′∈Wy. Therefore g♯(V)smapsx′/ma√sto→σx′(φ(c)) for allx′in the open neighbourhood g−1(Wy) ofxinU, by (4.8). Since the open subsets g−1(Wy) coverU,g♯(V)sis a section ofOX|U, andg♯(V) is well defined. As theφxareC∞-ring morphisms, this defines a morphism g♯:OY→ g∗(OX) of sheaves of C∞-rings onY. Letg♯:g−1(OY)→OXbe the corre- sponding morphism of sheaves on Xunder (4.3). The stalk g♯ x:OY,y→OX,x ofg♯atx∈Xwithg(x) =y∈Yisg♯ x=φx. Theng= (g,g♯) is a morphism inLC∞RS. SetLC,X(φ) =g. This defines LC,Xin (4.5). Forφ,gas above,c∈C, andx∈Xwithg(x) =y=x∗◦φ∈Y, we have σx/bracketleftbig/parenleftbig RC,X◦LC,X(φ)/parenrightbig (c)/bracketrightbig =σx/bracketleftbig Γ(g)◦ΨC(c)/bracketrightbig =g♯ x◦σy[ΨC(c)] =φx◦σy[ΨC(c)] =φx◦πy(c) =σx◦φ(c), usingLC,X(φ) =gand the definition (4.7) of RC,X(g) in the first step, σx◦ Γ(g) =g♯ x◦σy: Γ(Y)→OX,xin the second, g♯ x=φxin the third, σy◦ΨC=πy as maps C→OY,y∼=Cyin the fourth, and (4.8) in the fifth. As/producttext x∈Xσx: Γ(X)→/producttext x∈XOX,xis injective, this implies that/parenleftbig RC,X◦LC,X(φ)/parenrightbig (c) =φ(c) for allc∈C, soRC,X◦LC,X(φ) =φ, andRC,X◦LC,X= id. Supposef:X→Yis a morphism in LC∞RS, and setφ=RC,X(f) and g=LC,X(φ). Letx∈Xwithf(x) =y∈Y. Then we have a commutative diagram in C∞Rings C y /d38/d38◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆φ /d46/d46 πy/d15/d15ΨC/d47/d47Γ◦SpecC= Γ(Y) σy/d15/d15Γ(f)/d47/d47Γ(X) σx/d15/d15x∗ /d119/d119♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ Cy π/d43/d43❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲∼=/d47/d47OY,y π /d15/d15f♯ x/d47/d47OX,x π /d115/d115❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ R,(4.9) 29where the isomorphism Cy∼=OY,ycomes from Lemma 4.18. Since g(x) = x∗◦φ:C→R, this proves that g(x) =y=f(x), sof=g. Also by definition the stalkg♯ x:OY,y→OX,xisφxin (4.8), so comparing (4.8) and (4.9) and usingπy:C→Cysurjective by Proposition 2.14 shows that f♯ x=g♯ x. As this holds for all x∈Xwe havef♯=g♯, sof= (f,f♯) = (g,g♯) =g. Thus LC,X◦RC,X(f) =ffor allf:X→Y, soLC,X◦RC,X= id. Therefore LC,X,RC,Xin (4.5) are inverse bijections. It is easy to see that the rectangle in (4.6) involving RD,Y,RC,Xcommutes using (4.7) and functoriality of the Ψ Cand Γ. Then the rectangle involving LD,Y,LC,Xcommutes as LD,Y=R−1 D,YandLC,X=R−1 C,X. So (4.6) commutes. This completes the proof. Remark 4.21. (a) The fact in Theorem 4.20 that Spec : C∞Ringsop→ LC∞RSis right adjoint to Γ : LC∞RS→C∞Ringsopdetermines Spec uniquely up to natural isomorphism, by properties of adjoint funct ors. Dubuc [23] and Moerdijk, van Quˆ e and Reyes [52, §3] both prove the ex- istence of a right adjoint to Γ : LC∞RS→C∞Ringsop, which is therefore naturally isomorphic to our functor Spec in Definition 4.16. But they s how Spec exists by category theory, without constructing it explicitly as we d o. Moerdijk et al. [52, §3] call our functor Spec the Archimedean spectrum . They also give a nonequivalent definition [52, §1] of the spectrum Spec C, in which the points are not R-points, but ‘ C∞-radical prime ideals’. (b)Since Spec is a right adjoint functor, it preserves limits, as in [23, p. 687]. Equivalently, Spec takes colimits in C∞Ringsto limits in LC∞RS. So, for example, a pushout C=D∐FEof morphisms φ:F→D,ψ:F→Ein C∞Ringsis mapped to a fibre product Spec C∼=SpecD×SpecFSpecEof morphisms Spec φ: SpecD→SpecF, Specψ: SpecE→SpecFinLC∞RS. Here are some properties of finitely generated and fair C∞-rings, due to Dubuc [23, Th. 13]. The reflection functor Rfa fgis as in Definition 2.20. Theorem 4.22. (a) IfCis a finitely generated C∞-ring, there is a natural isomorphism Γ◦SpecC∼=Rfa fg(C),which identifies ΨC:C→Γ(SpecC)with the natural surjective projection C→Rfa fg(C). These isomorphisms for all Cform a natural isomorphism Rfa fg∼=Γ◦Spec of functors Rfa fg,Γ◦Spec :C∞Ringsfg→C∞Ringsfa. Hence, if Cis fair then ΨC:C→Γ(SpecC)∼=Rfa fg(C)is an isomorphism. (b)IfCis finitely generated then SpecΨ C: SpecC→SpecΓ(Spec C)∼= SpecRfa fg(C)is an isomorphism in LC∞RS. (c)The functor Spec|···: (C∞Ringsfa)op→LC∞RSis full and faithful, and takes finite limits in (C∞Ringsfa)opto finite limits in LC∞RS. To see that Spec is full and faithful on ( C∞Ringsfa)opin (c), let C,Dbe fairC∞-rings. Then putting X= SpecDin (4.5) and using D∼=Γ◦SpecDby (a) shows that the following is a bijection. Spec : Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC). 30Note that Spec is neither full nor faithful on ( C∞Ringsfg)oporC∞Ringsop. This is a contrast to conventional algebraic geometry, where Γ(Sp ecR)∼=Rfor arbitrary rings R, as in [31, Prop. II.2.2], so that Spec is full and faithful. In §4.6 we will generalize Theorem 4.22 to non-finitely-generated C∞-rings. 4.5 Affine C∞-schemes and C∞-schemes As for the usual definitions of affine schemes and schemes, we defin e: Definition 4.23. A localC∞-ringed space Xis called an affineC∞-scheme if it is isomorphic in LC∞RSto SpecCfor someC∞-ringC. We callXa finitely presented , orfair, affineC∞-scheme ifX∼=SpecCforCthat kind of C∞-ring. Write AC∞Sch,AC∞Schfp,AC∞Schfafor the full subcategories of affineC∞-schemes and of finitely presented, and fair, affine C∞-schemes in LC∞RSrespectively. We do not define finitely generated affineC∞-schemes, because Theorem 4.22(b) implies that they coincide with fair affine C∞-schemes. LetX= (X,OX) be a local C∞-ringed space. We call XaC∞-schemeif Xcan be covered by open sets U⊆Xsuch that ( U,OX|U) is an affine C∞- scheme. We call a C∞-schemeXlocally fair , orlocally finitely presented , ifX can be covered by open U⊆Xwith (U,OX|U) a fair, or finitely presented, affineC∞-scheme, respectively. We call aC∞-schemeXHausdorff ,second countable ,Lindel¨ of,compact, locally compact ,paracompact ,metrizable ,regular, orseparable , if the topological spaceXis. AffineC∞-schemes are Hausdorff and regular by Lemma 4.15. WriteC∞Schlf,C∞Schlfp,C∞Schfor the full subcategories in LC∞RS of locally fair C∞-schemes, locally finitely presented C∞-schemes, and all C∞- schemes, respectively. Remark 4.24. Ordinary schemes are a much larger class than ordinary affine schemes, and central examples such as CPnare not affine schemes. However, affineC∞-schemes are already general enough for many purposes. For ex ample, all second countable, metrizable C∞-schemes are affine, as in §4.8, including manifolds and manifolds with corners. Affine C∞-schemes are Hausdorff and regular, so any non-Hausdorff or non-regular C∞-scheme is not affine. For the next theorem, part (a) follows from Propositions 2.5, 2.24 a nd 2.26, Remark 4.21(b), and Theorem 4.22(c). Part (b) holds as finite lim- its inC∞Schlfp,C∞Schlf,C∞Schare locally modelled on finite limits in AC∞Schfp,AC∞SchfaandAC∞Sch. Theorem 4.25. (a) The full subcategories AC∞Schfp,AC∞Schfa,AC∞Sch are closed under all finite limits in LC∞RS. Hence, fibre products and all finite limits exist in each of these subcategories. (b)The full subcategories C∞Schlfp,C∞SchlfandC∞Schare closed under all finite limits in LC∞RS. Hence, fibre products and all finite limits exist in each of these subcategories. 31Definition 4.26. Define functors FC∞Sch Man:Man−→AC∞Schfp⊂AC∞Sch, FC∞Sch Manb:Manb−→AC∞Schfa⊂AC∞Sch, FC∞Sch Manc:Manc−→AC∞Schfa⊂AC∞Sch, byFC∞Sch Man∗= Spec◦FC∞Rings Man∗, in the notation of Definitions 3.2 and 4.16. By Example 4.17, if Xis a manifold with corners then FC∞Sch Manc(X) is nat- urally isomorphic to the local C∞-ringed space Xin Example 4.10. IfX,Y,... are manifolds, or f,g,...are (weakly) smooth maps, we may use X,Y,...,f,g,...to denote the images of X,Y,...,f,g,... underFC∞Sch Manc. So for instance we will write Rnand [0,∞)forFC∞Sch Man(Rn) andFC∞Sch Manb/parenleftbig [0,∞)/parenrightbig . Our categories of spaces so far are related as follows: Man FC∞Sch Man/d15/d15⊂/d47/d47Manb FC∞Sch Manb/d15/d15⊂/d47/d47Manc FC∞Sch Manc/d118/d118♥♥♥♥♥♥♥♥♥♥ AC∞Schfp ⊂/d47/d47 ⊂/d15/d15AC∞Schfa ⊂/d47/d47 ⊂/d15/d15AC∞Sch ⊂/d15/d15⊂ /d39/d39❖❖❖❖❖❖❖❖❖❖ C∞Schlfp⊂/d47/d47C∞Schlf⊂/d47/d47C∞Sch⊂/d47/d47LC∞RS⊂/d47/d47C∞RS. By Corollary 3.4 and Theorems 3.5 and 4.22(c), we find as in [23, Th. 16]: Corollary 4.27. FC∞Sch Man:Man֒→AC∞Schfp⊂AC∞Schis a full and faithful functor, and FC∞Sch Manb:Manb→AC∞Schfa⊂AC∞Sch, FC∞Sch Manc: Manc→AC∞Schfa⊂AC∞Schare both faithful functors, but are not full. Also these functors take transverse fibre products in Man,Mancto fibre prod- ucts inAC∞Schfp,AC∞Schfa. We study open subspaces of C∞-schemes. The definition of Spec Cimplies: Lemma 4.28. LetCbe aC∞-ring, andc∈C. WriteSpecC= (X,OX)and Uc={x∈X:x(c)/\e}atio\slash= 0}. ThenUc⊆Xis open with (Uc,OX|Uc)∼=SpecC[c−1]. Corollary 4.29. LetX= (X,OX)be aC∞-scheme and V⊆Xbe open. ThenV= (V,OX|V)is also aC∞-scheme. Proof.Letx∈V. Then there exists an open x∈Y⊆XwithY∼=SpecCfor someC∞-ringC, asXas aC∞-scheme. Identify Ywith Spec C. AsV∩Yis open inY=XC, and the topology on XCis generated by subsets Uc={˜x∈ XC: ˜x(c)/\e}atio\slash= 0}forc∈C, there exists c∈Csuch thatx∈Uc⊆V∩Y. Then (Uc,OX|Uc)∼=SpecC[c−1] by Lemma 4.28. So every x∈Vhas an affine open neighbourhood, and Vis aC∞-scheme. Lemma 4.30. LetCbe a finitely generated C∞-ring and (X,OX) = Spec C. SupposeV⊆Xis open. Then there exists c∈CwithV={x∈X:x(c)/\e}atio\slash= 0}. We callcacharacteristic function forV. 32Proof.AsCis a finitely generated C∞-ring it fits into an exact sequence 0 → I ֒→C∞(Rn)φ−→C→0. Example 4.14 gives a homeomorphism φ∗:X→Xφ C with a closed subset Xφ CinRngiven in (4.4). Then φ∗(V) is open in Xφ C, so there exists an open U⊆RnwithU∩Xφ C=φ∗(V). By [54, Lem. I.1.4] there existsf∈C∞(Rn) withU=/braceleftbig x∈Rn:f(x)/\e}atio\slash= 0/bracerightbig . Thenc=φ(f)∈Cis a characteristic function for V. Example 4.31. LetIbe an infinite set, and write C∞(RI) for the free C∞- ring with generators xifori∈I. ThenX= SpecC∞(RI) has topological space X=RIwith points ( xi)i∈Iforxi∈R. Elements of C∞(RI) are functions c:RI→Rdepending only on xjforjin afinitesubsetJ⊆I, and which are smooth functions of these xj,j∈J. LetV=RI\{0}. ThenVis open inX. But no characteristic function c exists forVinC∞(RI), sincecwould depend only on xjforjin a finite subset J⊆I, butVdepends on xifor alli∈I. Thus, infinitely generated C∞-rings need not admit characteristic functions, in contrast to Lemma 4.30 . IfCis a finitely generated (or finitely presented) C∞-ring andc∈Cthen C[c−1] is also finitely generated (or finitely presented), since C[c−1]∼=C[x]/(c· x−1) is the result of adding one extra generator and one extra relatio n toC. Thus from Lemmas 4.28 and 4.30 we deduce: Corollary 4.32. (a) Let(X,OX)be a fair (or finitely presented) affine C∞- scheme, and U⊆Xbe an open subset. Then (U,OX|U)is also a fair (or finitely presented) affine C∞-scheme. (b)Let(X,OX)be a locally fair (or locally finitely presented) C∞-scheme, and U⊆Xbe an open subset. Then (U,OX|U)is also a locally fair (or locally finitely presented) C∞-scheme. Our next result describes the sheaf of C∞-ringsOXin SpecCforCa finitely generatedC∞-ring. It is a version of [31, Prop. I.2.2(b)] in algebraic geometry, and reduces to Moerdijk and Reyes [54, Prop. I.1.6] when C=C∞(Rn). Proposition 4.33. LetCbe a finitely generated C∞-ring, write (X,OX) = SpecC,and letU⊆Xbe open. By Lemma 4.30we may choose a character- istic function c∈CforU. Then there is a canonical isomorphism OX(U)∼= Rfa fg(C[c−1]),in the notation of Definitions 2.13and2.20. IfCis finitely pre- sented thenOX(U)∼=C[c−1]. Proof.We have morphisms of C∞-ringsc∗:C∞(R)→Candi∗:C∞(R)→ C∞(R\{0}),andC∞(R),C∞(R\{0})arefinitelypresented C∞-ringsbyPropo- sition 3.1(a). So as Spec preserves limits in ( C∞Ringsfg)opwe have Spec/parenleftbig C∐c∗,C∞(R),i∗C∞(R\{0})/parenrightbig∼=SpecC×f,R,iR\{0}∼=(U,OX|U). ButC∐C∞(R)C∞(R\{0})∼=C[c−1] for formal reasons. Thus Theorem 4.22(a) givesOX(U)∼=Γ/parenleftbig (U,OX|U)/parenrightbig∼=Rfa fg(C[c−1]). IfCis finitely presented then C[c−1] is too, as in Corollary 4.32, so C[c−1] is fair and Rfa fg/parenleftbig C[c−1]/parenrightbig =C[c−1], and thereforeOX(U)∼=C[c−1]. 334.6 Complete C∞-rings The material of this section appears to be new. Proposition 4.34. LetCbe aC∞-ring, and ΨCbe as in Definition 4.19. Then SpecΨ C: Spec◦Γ◦SpecC→SpecCis an isomorphism in LC∞RS. Proof.WriteD= Γ◦SpecC,X= SpecC,Y= SpecD, andf= SpecΨ C:Y→ X. Letx∈X, and define y=π◦Πx:D→Rto be the composition of the projection Π x:D→Cx, noting that D⊆/producttext ˜x∈XC˜xby Definition 4.19, and the unique morphism π:Cx→R, asCxis a localC∞-ring. Then f(y) =π◦πx= x:C→Rforπx:C→Cx, sof:Y→Xis surjective. Suppose now that y∈Ywithf(y) =x, so thaty:D→Ris anR-algebra morphism. We will prove that y=π◦Πxas above. Let d∈D. By definition of D=OXC(XC) there exist an open neighbourhood WofxinXandc1∈Csuch thatd(˜x) =π˜x(c1) inC˜xfor all ˜x∈W. By definition of the topology TC, there existsc2∈Csuch thatUc2={˜x∈X: ˜x(c2)/\e}atio\slash= 0}is an open neighbourhood of xinW⊆X. Hencex(c2)/\e}atio\slash= 0 and ˜x(c2) = 0 for all ˜ x∈X\W. Choose smooth functions g,h:R→Rwithg(x(c2)) = 1 and g= 0 in an open neighbourhood ( −ǫ,ǫ) of 0 in R, andh(0)/\e}atio\slash= 0 andh= 0 outside (−ǫ,ǫ), so thatg·h= 0. Setc3= Φg(c2) andc4= Φh(c2), with Φ g,Φh:C→Cthe C∞-ring operations. Then x(c3) = 1, andπ˜x(c3) = 0 inC˜xfor all ˜x∈X\W, as π˜x(c3)·π˜x(c4) =π˜x/parenleftbig Φg(c2)·Φh(c2)/parenrightbig =π˜x◦Φgh(c2) =π˜x◦Φ0(c2) = 0, butπ˜x(c4) is invertible in C˜xas ˜x(c4) =h(˜x(c2)) =h(0)/\e}atio\slash= 0. Thus we have d·ΨC(c3) = ΨC(c1)·ΨC(c3) = ΨC(c1·c3) inD, asd(˜x) = ΨC(c1)˜xfor all ˜x∈W, and Ψ C(c3)˜x= 0 for all ˜x∈X\W. Therefore y(d) =y(d)·1 =y(d)·x(c3) =y(d)·y(ΨC(c3)) =y/parenleftbig d·ΨC(c3)/parenrightbig =y/parenleftbig ΨC(c1·c3)/parenrightbig =x(c1·c3)=x(c1)·x(c3)=/parenleftbig π◦Πx(d)/parenrightbig ·1=π◦Πx(d). As this holds for all d∈D, we see that y∈Ywithf(y) =ximplies that y=π◦Πx. Hencef:Y→Xis injective, and so bijective. From above f:Y→Xis continuous. To show f−1:X→Yis continuous, note that the topology on Yis generated by the basis of open sets Vd={y∈ Y:y(d)/\e}atio\slash= 0}for alld∈D. So it is enough to show that f(Vd) ={x∈X: π◦Πx(d) = 0}is open inXfor alld. For fixed d, by definition we may cover Xby openW⊆Xfor which there exist c∈Cwithd(x) =πx(c)∈Cxfor all x∈W. But then W∩f(Vd) =W∩Uc, whereUc={x∈X:x(c)/\e}atio\slash= 0}is open inX. So we can cover Xby openW⊆XwithW∩f(Vd) open, and f(Vd) is open. Therefore f−1is continuous, and f:Y→Xis a homeomorphism. Lety∈Ywithf(y) =x. Taking stalks of f♯:f−1(OX)→OYatygives a morphismf♯ y:OX,x→OY,y, whereOX,x∼=CxandOY,y∼=Dyby Lemma 4.18, and we have a commutative diagram C πx/d15/d15ΨC/d47/d47D πy/d15/d15Πx/d114/d114❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ Cx∼=OX,xΨC,x∼=f♯ y/d47/d47OY,y∼=Dy.(4.10) 34Here the outer rectangle and top left triangle obviously commute. T o see that the bottom right triangle commutes, we use that any d∈D=OXC(XC) has d(˜x) = Ψ C(c)˜xfor somec∈Cand all ˜xin an open neighbourhood Wofxin X. As in the first part of the proof, we can find c3∈Cwithx(c3) = 1 and π˜x(c3) = 0 inC˜xfor all ˜x∈X\W. Then evaluating at ˜ x∈Wand ˜x∈X\W we see that Ψ C(c)·ΨC(c3) =d·ΨC(c3), which forces πy(d) =πy(ΨC(c)), since πy◦ΨC(c3) is invertible in Dyasπ◦πy◦ΨC(c3) =x(c3) = 1>0. Thus πy(d) =πy◦ΨC(c) =f♯ y◦πx(c) =f♯ y◦Πx◦ΨC(c) =f♯ y◦Πx(d). Sinceπy:D→Dyis surjective by Proposition 2.14, the bottom right trianglein(4.10)impliesthat f♯ y:OX,x→OY,yissurjective. Suppose cx∈OX,x withf♯ y(cx) = 0 inOY,y. Asπxis surjective by Proposition 2.14 we may writecx=πx(c) forc∈C. Thenπy◦ΨC(c) =f♯ y◦πx(c) =f♯ y(cx) = 0, so ΨC(c)∈Kerπy. WriteI⊂CandJ⊂Dfor the ideals in (2.2) for x,y. Then J= Kerπy, so ΨC(c)∈J, and thus there exists d∈Dwithy(d) =π◦Πx(d)/\e}atio\slash= 0 inRand Ψ C(c)·d= 0 inD. Applying Π xgives cx·Πx(d) =πx(c)·Πx(d) = Πx(ΨC(c))·Πx(d) = Πx(ΨC(c)·d) = Πx(0) = 0. But Πx(d) is invertible in Cxasπ◦Πx(d)/\e}atio\slash= 0 inR, socx= 0. Thusf♯ y:OX,x→ OY,yis injective, and so an isomorphism. We have shown that f:Y→Xis a homeomorphism, and f♯ y:OX,f(y)→ OY,yis an isomorphism on stalks at all y∈Y. Hence SpecΨ C= (f,f♯) is an isomorphism in LC∞RS, as we have to prove. Definition 4.35. We call aC∞-ringCcomplete if the morphism Ψ C:C→ Γ◦SpecCin Definition 4.19 is an isomorphism. Write C∞Ringscofor the full subcategory of complete C∞-ringsCinC∞Rings. IfCis anyC∞-ring, applying Γ to SpecΨ Cin Proposition 4.34 shows that Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC) is an isomorphism in C∞Rings, where we check that Γ ◦SpecΨ C= ΨΓ◦SpecC from Definitions 4.16 and 4.19. Hence Γ ◦SpecCis a complete C∞-ring. Define a functorRco all:C∞Rings→C∞RingscobyRco all= Γ◦Spec. The next result extends Definition 2.20 and Theorem 4.22 from C∞Ringsfa ⊂C∞RingsfgtoC∞Ringsco⊂C∞Rings. Theorem 4.36. (a) LetXbe an affine C∞-scheme. Then X∼=SpecOX(X), whereOX(X)is a complete C∞-ring. (b)Spec|(C∞Ringsco)op: (C∞Ringsco)op→LC∞RSis full and faithful, and an equivalence of categories Spec|···: (C∞Ringsco)op→AC∞Sch. (c)Rco all:C∞Rings→C∞Ringscois left adjoint to the inclusion functor inc :C∞Ringsco֒→C∞Rings. That is,Rco allis areflection functor . (d)All small colimits exist in C∞Ringsco,although they may not coincide with the corresponding small colimits in C∞Rings. 35(e)Spec|(C∞Ringsco)op= Spec◦inc : (C∞Ringsco)op→LC∞RSis right adjoint toRco all◦Γ :LC∞RS→(C∞Ringsco)op. ThusSpec|···takes limits in (C∞Ringsco)op(equivalently, colimits in C∞Ringsco) to limits in LC∞RS. Proof.For (a), ifXis an affine C∞-scheme then X∼=SpecCfor someC∞-ring C, soOX(X)∼=Γ◦SpecC, and thus X∼=SpecOX(X) by Proposition 4.34. Also, applying Γ to SpecΨ Cin Proposition 4.34 shows that Γ◦SpecΨ C= ΨΓ◦SpecC: Γ◦SpecC−→Γ◦Spec(Γ◦SpecC) is an isomorphism in C∞Rings, where Γ◦SpecΨ C= ΨΓ◦SpecCfollows from the definitions. Hence Γ ◦SpecC∼=OX(X) is complete, proving (a). For (b), if C,Dare complete C∞-ringsthen putting X= SpecDin Theorem 4.20 and using Γ ◦SpecD∼=D, equation (4.5) shows that Spec =LC,X: Hom C∞Rings(C,D)−→HomLC∞RS(SpecD,SpecC) is a bijection, where the definition of LC,Xagrees with the definition of Spec on morphisms in this case. Thus Spec is full and faithful on complete C∞-rings. Therefore Spec|···: (C∞Ringsco)op→LC∞RSis an equivalence of categories from (C∞Ringsco)opto its essential image in LC∞RS, which is AC∞Sch. For (c), let C,DbeC∞-rings with Dcomplete. Then we have bijections HomC∞Ringsco/parenleftbig Rco all(C),D/parenrightbig∼=HomC∞Rings/parenleftbig Γ◦SpecC,Γ◦SpecD/parenrightbig ∼=HomLC∞RS/parenleftbig SpecD,Spec◦Γ◦SpecC/parenrightbig∼=HomLC∞RS/parenleftbig SpecD,SpecC/parenrightbig ∼=HomC∞Rings/parenleftbig C,Γ◦SpecD/parenrightbig∼=HomC∞Rings/parenleftbig C,D/parenrightbig = Hom C∞Rings/parenleftbig C,inc(D)/parenrightbig , (4.11) usingD∼=Γ◦SpecDasDis complete in the first and fifth steps, Theorem 4.20 in the second and fourth, and Proposition 4.34 in the third. The b ijections (4.11) are functorial in C,Das each step is. Hence Rco allis left adjoint to inc. For (d), note that Rco all:C∞Rings→C∞Ringscotakes colimits to colim- its, asit isaleft adjointfunctor by(a). Sogivenafunctor F:J→C∞Ringsco forJa small category, we may take the colimit C= colim JFinC∞Rings, which exists by Proposition 2.5, and then D=Rco all(C) is the colimit of Rco all◦F inC∞Ringsco. ButRco all◦F∼=FasRco all|C∞Ringsco∼=id. Hence D= colim JF inC∞Ringsco, and all small colimits exist in C∞Ringsco. In Example 2.25, the colimits in C∞RingscoandC∞Ringsare different. The first part of (e) holds by composing (c) and Theorem 4.20, and t he second part follows as right adjoint functors preserve limits. This c ompletes the proof of Theorem 4.36. Remark 4.37. LetCbe aC∞-ring, so that Ψ C:C→Rco all(C) is a morphism of C∞-rings. If Cis finitely generated then Theorem 4.22(a) gives an isomorphism Rco all(C)∼=Rfa fg(C) identifying Ψ Cwith the surjective projection π:C→Rfa fg(C), forRfa fgas in Definition 2.20. Thus Ψ C:C→Rco all(C) is surjective in this case, andRco all,Rfa fgagree on finitely generated C∞-rings up to natural isomorphism. 36ForCinfinitely generated, Ψ C:C→Rco all(C) need not be surjective, and Rco all(C) can be much larger than C. For example, if Iis an infinite set and C=C∞(RI) is as in Example 4.31, then elements of Care functions c:RI→R which depend smoothly only on xjforjin a finite subset J⊆I, but elements ofRco all(C) are functions c:RI→Rwhichlocally in RIdepend smoothly only onxjforjin a finite subset J⊆I, but globally may depend on xifor infinitely manyi∈I. So Ψ C:C→Rco all(C) is injective but not surjective. 4.7 Partitions of unity We now study the existence of smooth partitions on unity on C∞-schemes and localC∞-ringed spaces. We will need the next definition. Definition 4.38. LetX= (X,OX) be a local C∞-ringed space. Then each c∈OX(X) defines a continuous map c∗:X→Rmappingx/ma√sto→π◦πx(c), for πx:OX(X)→OX,xandπ:OX,x→Rthe natural C∞-ring morphisms. Thus Uc={x∈X:c∗(x)/\e}atio\slash= 0}is open inX. We say that the topology on Xis smoothly generated if{Uc:c∈OX(X)}is a basis for the topology on X. This implies Xis a regular (and completely regular) topological space. Example 4.39. (a) LetXbeacompletelyregulartopologicalspace, anddefine a sheaf ofC∞-ringsOXonXby takingOX(U) =C0(U) to be the C∞-ring of continuous functions c:U→Rfor all open U⊆X. ThenX= (X,OX) is a localC∞-ringed space, and the topology on Xis smoothly generated. (b)LetXbe an affine C∞-scheme. Then X∼=SpecOX(X) by Theorem 4.36(a). So the definition of the topology on Xin Definition 4.13 implies that the topology on Xis smoothly generated. (c)SupposeXis a regular C∞-scheme, and let T⊆Xbe open and x∈T. Thenxhas an affine open neighbourhood YinX. SinceXis regular, there exist disjoint open neighbourhoods VofxandWofX\YinX. Thenx∈T∩V⊆Y, and the topology on Yis smoothly generated by (b), so there exists a∈OY(Y) withx∈UY a⊆T∩V. Nowa∗(x)/\e}atio\slash= 0 anda∗(y) = 0 for ally∈Y\UY a, but this does not imply that ais supported in UY a, as we could have πy(a)/\e}atio\slash= 0 inOY,yeven though π◦πy(a) = 0 in R. Choose smooth f:R→Rwithf(a∗(x))/\e}atio\slash= 0 andf(t) = 0 fortin an open neighbourhood of 0 inR. Setb= Φf(a), for Φf:OY(Y)→OY(Y) theC∞-ring operation. Thenb∗(x)/\e}atio\slash= 0, andUY b⊆UY a⊆T, andbis supported in UY a⊆V⊆Y. SinceWis open inXwithX\Y⊆W⊆Y\V, there exists a unique c∈OX(X) withc|Y=bandc|W= 0. We have x∈UX c=UY b⊆T. Thus, for each open T⊆Xandx∈Twe can find c∈OX(X) withx∈UX c⊆T. So the topology onXis smoothly generated. (d)LetXbe an infinite-dimensional Banach space or Banach manifold, and makeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10. The question of when the topology of Xis smoothly generated (framed in terms of the existence of ‘smooth bump functions’ on X) is very well understood, as in Bonic and Frampton [10] and Deville, Godefroy and Zizler [18, §V]. For 37example, if Xis a Hilbert manifold, or modelled on Lq(Y) orℓqfor evenq/greaterorequalslant2, then the topology on Xis smoothly generated, but if Xis modelled on Lq(Y) orℓqforq∈[1,∞] not even, the topology on Xis not smoothly generated. For the next theorem, §4.1 defined Lindel¨ of spaces, and explained their rela- tion to other topological assumptions. Second countable implies Lind el¨ of, and Lindel¨ of and regular imply paracompact (note that Xis regular as its topology is smoothly generated). It is easy to see that OXfine implies that the topology onXis smoothly generated. The proof of Theorem 4.40 is based on the proof of the existence of smooth partitions on unity on suitable separableBanachmanifolds in Bonic and Framp- ton [10, Th. 1] (see also Lang [45, §II.3] and Deville et al. [18, §VIII.3]). Theorem 4.40 applies to a very large class of C∞-schemes, showing that partitions of unity exist on most interesting examples of C∞-schemes. Theorem 4.40. LetX= (X,OX)be a Lindel¨ of local C∞-ringed space, and suppose the topology on Xis smoothly generated. Then OXisfine, as in Definition 4.7. That is, for every open cover {Vi:i∈I}ofXthere exists a subordinate locally finite partition of unity {ηi:i∈I}inOX(X). Proof.Forc∈OX(X) andx∈Xwe haveπx(c)∈OX,xandc∗(x) =π◦ πx(c)∈R, whereπx:OX(X)→OX,xandπ:OX,x→Rare the natural C∞- morphisms. Then c∗:X→Ris continuous. Write Uc={x∈X:c∗(x)/\e}atio\slash= 0}, so thatUcis open inX. Thesupportofcis suppc={x∈X:πx(c)/\e}atio\slash= 0}. Then suppcis closed in XwithUc⊆suppc, but suppcmay be larger than the closure of Uc. Note that an infinite sum/summationtext j∈JcjinOX(X) is defined, as a section of the sheaf OX, if{suppcj:j∈J}is locally finite (that is, each x∈Xhas an open neighbourhood Wxintersecting supp cjfor only finitely manyj∈J), but may not make sense if only {Ucj:j∈J}is locally finite. Because of this, we are careful to keep track of both Ucjand suppcjin the following proof. Let{Vi:i∈I}be an open cover of X. Supposei∈Iandx∈Vi. As the topology on Xis smoothly generated there exists c∈OX(X) withx∈Uc⊆Vi. Soc∗(x)/\e}atio\slash= 0 andc∗|X\Vi= 0. We do not know that supp c⊆Vi, but we can correct this as follows. Choose smooth f:R→Rsuch thatf(c∗(x))/\e}atio\slash= 0 and f= 0 in a neighbourhood of 0 in R. Setc′= Φf(c), where Φ f:OX(X)→ OX(X) is theC∞-ring operation. Then x∈Uc′⊆suppc′⊆Uc⊆Vi⊆X. Thus, we can choose a family {cj:j∈J}such thatcj∈OX(X), and Ucj⊆suppcj⊆Vij⊆Xfor eachj∈Jand someij∈I, and{Ucj:j∈J} is an open cover of X. SinceXis Lindel¨ of we can take Jto be countable, and chooseJ=N. Replacingcjbyc2 jwe have (cj)∗/greaterorequalslant0 onX. For eachj∈N, choose smooth fj:Rj+1→Rsuch thatfj(t0,t1,...,tj)>0 ifti<1/jfori= 0,1,...,j−1 andtj>0, andfj(t0,t1,...,tj) = 0 otherwise. Define dj= Φfj(c0,c1,...,cj), 38with Φfj:OX(X)j+1→OX(X) theC∞-ring operation. Then Udj=/braceleftbig x∈X: (dj)∗(x)/\e}atio\slash= 0/bracerightbig =/braceleftbig x∈X: (ci)∗(x)<1/j, i= 0,...,j−1,(cj)∗(x)/\e}atio\slash= 0/bracerightbig ⊆Vij, suppdj⊆/braceleftbig x∈X: (ci)∗(x)/lessorequalslant1/j, i= 1,...,j−1/bracerightbig ∩suppcj⊆Vij.(4.12) Fixx∈X. Thenx∈Ucjfor somej∈Nas{Ucj:j∈J}coversX. Letj∈Nbe least with x∈Ucj. Then (cj)∗(x)>0 and (ci)∗(x) = 0 for i= 0,1,...,j−1. Thusx∈Udj, so{Udj:j∈N}is an open cover of X. Define Tx={y∈X: (cj)∗(y)>1 2(cj)∗(x)}. ThenTxis an open neighbourhood of x inX, andTx∩Udk=∅=Tx∩suppdkprovidedk >max/parenleftbig j,2(cj)∗(x)−1/parenrightbig by (4.12). Thus, both {Udj:j∈N}and{suppdj:j∈N}are locally finite. For eachi∈I, defineei=/summationtext j∈N:ij=idjinOX(X). This is well defined as{suppdj:j∈N}is locally finite. We have Uei⊆suppei⊆Vi, since Udj⊆suppdj⊆Vifor eachj∈Nwithij=i. Both{Uei:i∈I}and {suppei:i∈I}are locally finite, as {Udj:j∈N}and{suppdj:j∈N}are. Thuse=/summationtext i∈Ieiis well defined in OX(X). Ifx∈Xthen e∗(x) =/summationtext i∈I(ei)∗(x) =/summationtext i∈I/summationtext j∈N:ij=i(dj)∗(x) =/summationtext j∈N(dj)∗(x)>0, where each sum has only finitely many nonzero terms, and/summationtext j∈N(dj)∗(x)>0 as {Udj:j∈N}coversXwith (dj)∗>0 onUdjand (dj)∗= 0 onX\Udj. Since e∗is positive on X,eis invertible inOX(X). Setηi=e−1·eifori∈I. Then suppηi⊆Vi, assuppei⊆Vi, and{ηi:i∈I}islocallyfinite, as {suppei:i∈I} is, and/summationtext i∈Iηi=/summationtext i∈Ie−1·ei=e−1·e= 1. Hence{ηi:i∈I}is a locally finite partition of unity subordinate to {Vi:i∈I}, soOXis fine. 4.8 A criterion for affine C∞-schemes Here are sufficient conditions for a local C∞-ringed space Xto be an affine C∞- scheme. Note that affine C∞-schemes are Hausdorff with smoothly generated topology by Lemma 4.15 and Example 4.39(b), so Lindel¨ of is the only co ndition in the theorem which is not also necessary. Theorem 4.41. LetX= (X,OX)be a Hausdorff, Lindel¨ of, local C∞-ringed space, with smoothly generated topology. Then Xis an affine C∞-scheme. Proof.LetXbe as in the theorem. Note that Theorem 4.40 shows that OX is fine. Write C=OX(X) = Γ(X), andY= SpecC. Define a morphism f:X→Ybyf=LC,X(idC), using the notation of Theorem 4.20. We will showfis an isomorphism, so that X∼=SpecCis an affine C∞-scheme. Pointsx∈XinduceC∞-ring morphisms π◦πx:C=OX(X)→R, where πx:OX(X)→OX,xandπ:OX,x→Rare the natural projections. Points y∈YareC∞-ring morphisms y:C→R, andf:X→Yisf(x) =π◦πx. Supposex,x′∈Xwithx/\e}atio\slash=x′, and setf(x) =yandf(x′) =y′. SinceXis Hausdorff there exists open U⊆Xwithx∈Uandx′/∈U. As the topology on Xissmoothlygeneratedthereexists c∈OX(X)withc∗(x)/\e}atio\slash= 0andc∗|X\U= 0, 39so thatc∗(x′) = 0. Then y(c) =c∗(x)/\e}atio\slash= 0 andy′(c) =c∗(x′) = 0, soy/\e}atio\slash=y′. Hencef:X→Yis injective. Suppose for a contradiction that y∈Y, butf(x)/\e}atio\slash=yfor allx∈X. Then for eachx∈X, there exists a∈Cwithy(a)/\e}atio\slash=π◦πx(a). Choose smooth g:R→Rwithg(y(a)) = 0 andg= 1 in an open neighbourhood of π◦πx(a) in R. Setb= Φg(a), where Φ g:C→Cis theC∞-ring operation. Then y(b) = 0 andπ◦π˜x(b) = 1 for ˜xin an open neighbourhood VofxinX. Thus we may choose a family of pairs {(Vj,bj) :j∈J}such that for each j∈Jwe haveVj⊆Xopen andbj∈Cwithy(bj) = 0 andπ◦πx(bj) = 1 for x∈Vj, and{Vj:j∈J}is an open cover of X. AsXis Lindel¨ of we can suppose Jis countable, and so take J=N. By Theorem 4.40 there exists a locally finite partition of unity {ηj:j∈N}inCsubordinate to{Vj:j∈N}. Setc=/summationtext j∈Nj·ηj·bjinC=OX(X), which makes sense in global sections ofOXas{ηj:j∈N}is locally finite. Choose n∈Nwithn>y(c), and define d=c−y(c)·1X+/summationtextn−1 j=0(n−j)·ηj·bjinC, where 1X∈Cis the identity. Then y(d) =y(c)−y(c)·y(1X)+/summationtextn−1 j=0(n−j)·y(ηj)·y(bj) = 0, asy(1X) = 1 andy(bj) = 0. And if x∈Xthen π◦πx(d) =π◦πx/bracketleftbig/summationtext j∈Nj·ηj·bj−y(c)·/summationtext j∈Nηj+/summationtextn−1 j=0(n−j)·ηj·bj/bracketrightbig =/summationtext j∈N/parenleftbig max(j,n)−y(c)/parenrightbig π◦πx(ηj)>0, whereeachsumhasonlyfinitelymanynonzeroterms,andweuse/summationtext j∈Nηj= 1X, π◦πx(bj) = 1, and max( j,n)−y(c)>0,π◦πx(ηj)/greaterorequalslant0 forj∈N. Sinceπ◦πx(d)>0 for allx∈X, we see that dis invertible in C=OX(X), but this contradicts y(d) = 0. Hence each y∈Yhasy=f(x) for somex∈X, andfis surjective, so f:X→Yis a bijection. By definition of Y= SpecC, the topology on Yis generated by the open sets Uc={y∈Y:y(c)/\e}atio\slash= 0}for allc∈C. As the topology on Xis smoothly generated, it is generated by the open setsf−1(Uc) ={x∈X:c∗(x)/\e}atio\slash= 0}forc∈C. Therefore f:X→Yis a bijection identifying bases for the topologies of X,Y, sofis a homeomorphism. Letx∈Xwithf(x) =y∈Y. Taking stalks of f♯:f−1(OY)→OXatx gives a morphism f♯ x:OY,y→OX,x. By the definition of f=LC,X(idC) in the proof of Theorem 4.20, f♯ xagrees with φxin (4.8), and is the unique morphism making the following commute, where Cy∼=OY,yby Lemma 4.18: C πy/d15/d15y /d34/d34OX(X) πx/d15/d15 Cy∼=OY,y π/d42/d42❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱f♯ x/d47/d47OX,x π/d116/d116✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ R.(4.13) Supposeay∈OY,ywithf♯ x(ay) = 0. Then ay=πy(a) for some a∈C= OX(X), asπyis surjective by Proposition 2.14, and then πx(a) = 0 inOX,x, as 40(4.13) commutes. Hence there exists an open neighbourhood UofxinXwith a|U= 0 inOX(U). As the topology on Xis smoothly generated, there exists b∈OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth g:R→Rwith g(b∗(x))/\e}atio\slash= 0 andg= 0 near 0 in R, and setc= Φg(b), where Φ g:OX(X)→ OX(X) is theC∞-ring operation. Then y(c) =c∗(x)/\e}atio\slash= 0, andcis supported in U. Asa|U= 0 we see that a·c= 0 inOX(X). Thusalies in the ideal Iin (2.2) which is the kernel of πy:C→Cy, by Proposition 2.14, and so ay=πy(a) = 0. Thereforef♯ x:OY,y→OX,xis injective. Supposeax∈OX,x. Thenby definition of OX,xthereexists open x∈U⊆X anda∈OX(U) withπx(a) =ax. As the topology on Xis smoothly generated there exists b∈ OX(X) withb∗(x)/\e}atio\slash= 0 andb∗|X\U= 0. Choose smooth g:R→Rwithg= 1 nearb∗(x) inRandg= 0 near 0 in R. Setc= Φg(b), where Φg:OX(X)→OX(X) is theC∞-ring operation. Then cis supported inU, and there exists an open neighbourhood VofxinUwithc|V= 1. Since cis supported in U, the section c|U·a∈OX(U) can be extended by zero over X\Uto give a unique d∈OX(X) supported in Uwithd|U=c|U·a. Thend|V=c|V·a|V= 1·a|V=a|V. Hencef♯ x◦πy(d) =πx(d) =ax, sof♯ x:OY,y→ OX,xis surjective, and an isomorphism. This proves that f♯:f−1(OY)→OXis an isomorphism on stalks at every x∈X, sof♯is an isomorphism. As fis a homeomorphism, f= (f,f♯) :X→SpecCis an isomorphism. This completes the proof of Theorem 4.41. Corollary 4.42. LetX= (X,OX)be a localC∞-ringed space. Then the following are equivalent: (i)Xis Hausdorff and second countable, with smoothly generated t opology. (ii)Xis separable and metrizable, with smoothly generated topol ogy. (iii)Xis a Hausdorff, second countable, regular C∞-scheme. (iv)Xis a separable, metrizable C∞-scheme. (v)Xis a second countable, affine C∞-scheme. When these hold, Xis regular, normal, and paracompact, and OXis fine. Proof.Section 4.1 implies that (i),(ii) are equivalent (as Xsmoothly generated topology implies Xregular), and (iii),(iv) are equivalent. Also (v) implies (iii) by Lemma 4.15, and (iii) implies (i) by Example 4.39(b), and (i) implies (v) by Theorem 4.41 (as second countable implies Lindel¨ of). Hence (i)–( v) are equivalent. The last part follows from §4.1 and Theorem 4.40. In comparison to Theorem 4.41, we have strengthened the Lindel¨ o f assump- tion to second countable. The category of C∞-schemes in Corollary 4.42 is very large, and convenient to work in. They are closed under products, fibre prod- ucts, and arbitrary subspaces (Lindel¨ of spaces are none of the se). They have partitions of unity, and as they are affine we can argue globally using C∞-rings. 41Example 4.43. LetX= (X,OX) be a second countable, affine C∞-scheme, and letY⊆Xbeanysubset, not necessarily open or closed. Then Y= (Y,OX|Y) is also a second countable, affine C∞-scheme by Corollary 4.42, as being Hausdorff, second countable, and of smoothly generated to pology, are all preserved under passing to subspaces, so Ysatisfies Corollary4.42(i) as Xdoes. Example 4.44. LetXbe a separable Banach manifold modelled locally on separableBanach spaces Bwhich admit ‘smooth bump functions’ (that is, there exists a nonzero smooth function f:B→Rwith bounded support in B). See Deville et al. [18, §V] for results on when a Banach space Bhas a smooth bump function, for example, every Hilbert space does. MakeXinto a local C∞-ringed space X= (X,OX) as in Example 4.10. Then the topology on Xis smoothly generated as in Example 4.39(d), so Xis an affineC∞-scheme by Corollary 4.42(ii),(v). 4.9 Quotients of C∞-schemes by finite groups Finally we discuss quotients of C∞-schemes by finite groups. Definition 4.45. LetX= (X,OX) be a local C∞-ringed space, Ga finite group, and r:G→Aut(X) an action of GonX. We will define a local C∞-ringed space Y=X/G. SetY=X/r(G) to be the quotient topological space. Open sets V⊆Yare of the form U/GforU⊆Xopen andG-invariant. Then γ/ma√sto→r♯(γ)(U) gives an action ofGon theC∞-ringOX(U), so as in Proposition2.22we havea C∞-ring OX(U)G, theG-invariant subspace in OX(U). DefineOY(V) =OX(U)G. IfV2⊆V1⊆Yare open then V1=U1/G,V2=U2/GforU2⊆U1⊆X open andG-invariant. The restriction morphism ρU1U2:OX(U1)→OX(U2) in OXisG-equivariant, and so restricts to ρU1U2|OX(U1)G:OX(U1)G→OX(U2)G. SetρV1V2=ρU1U2|OX(U1)G:OY(V1)→OY(V2). It is now easy to check that OYis a sheaf of C∞-rings onY, soY= (Y,OY) is aC∞-ringed space. Ifx∈Xandy=xG∈Y, the stalkOY,yofOYatyis (OX,x)H, where OX,xis a localC∞-ring, andH=/braceleftbig γ∈G:γ(x) =x/bracerightbig is the stabilizer group ofxinG, which acts onOX,xin the obvious way. As OX,xis local there is anR-algebra morphism π:OX,x→R, such that c∈OX,xis invertible if and only ifπ(c)/\e}atio\slash= 0. Thusπ|(OX,x)H: (OX,x)H→Ris anR-algebra morphism, and c∈(OX,x)His invertible inOX,xif and only if π(c)/\e}atio\slash= 0. But if c∈(OX,x)H is invertible inOX,xthenc−1isH-invariant, so cis invertible in (OX,x)H. ThereforeOY,y∼=(OX,x)His a localC∞-ring, and Yis a localC∞-ringed space. Write X/G=Y. Defineπ:X→X/Gto be the natural projection. Define a morphism π♯:OY→π∗(OX) of sheaves of C∞-rings onY=X/Gby π♯(V) = inc :OY(V) =OX(U)G−→OX(U) =π∗(OX)(V) for all open V=U/G⊆Y=X/G, where inc :OX(U)G֒→OX(U) is the inclusion. Let π♯:π−1(OY)→OXbe the morphism of sheaves of C∞-rings on 42Xcorrespondingto π♯under (4.3). Then π= (π,π♯) :X→X/Gis a morphism of localC∞-ringed spaces. It is easy to see that X/G,πhave the universal property that if f:X→Z is a morphism in LC∞RSwithf◦r(γ) =ffor allγ∈Gthenf=g◦πfor a unique morphism g:X/G→ZinLC∞RS. Proposition 4.46. LetX= (X,OX)be an affine C∞-scheme,Ga finite group, and r:G→Aut(X)an action of GonX. SupposeXis Lindel¨ of. ThenX= SpecCforC=OX(X)a complete C∞-ring, andr= Specsfor s:G→Aut(C)a unique action of GonC. Form the G-invariantC∞-ring CG⊆Cas in Proposition 2.22. ThenCGis complete, and there is a canonical isomorphism X/G∼=SpecCGinLC∞RS. Proof.Theorem 4.36(a) shows that X∼=SpecC, whereC=OX(X) is a com- pleteC∞-ring. As Spec is full and faithful on complete C∞-rings by Theorem 4.36(b), Spec : Aut( C)→Aut(X) is an isomorphism, so there is a unique action s:G→Aut(C) withr= Specs. LetY=X/Gbe as in Definition 4.45. Then Y=X/Gis Hausdorff, as X is Hausdorff and Gis finite. Suppose {Vi:i∈I}is an open cover of Y. Then Vi=Ui/Gfor{Ui:i∈I}an open cover of X. AsXis Lindel¨ of there exists a subcover{Ui:i∈S}for countable S⊆I, and then{Vi:i∈S}is a countable subcover of{Vi:i∈I}. HenceYis Lindel¨ of. SupposeV⊆Yis open and y∈V. ThenV=U/Gandy=xGforG- invariantopen U⊆Xwithx∈U. As the topologyon Xis smoothly generated, there exists c∈Cwithc∗(x)/\e}atio\slash= 0 andc∗(x′) = 0 for all x′∈X\U. Define d=/summationtext γ∈Gγ∗(c2) inC. ThendisG-invariant with d∗(x)>0 andd∗(x′) = 0 for allx′∈X\U. Henced∈OY(Y) =OX(X)G=CG, withd∗(y)>0 and d∗(y′) = 0 for all y′∈Y\V. Thus the topology of Yis smoothly generated. Theorem 4.41 now implies that Y=X/Gis an affine C∞-scheme, and Theorem4.36(a)givesacanonicalisomorphism X/G∼=SpecOY(Y) = Spec CG, whereCGis complete. Proposition 4.47. SupposeXis a Hausdorff, second countable C∞-scheme, Ga finite group, and r:G→Aut(X)an action of GonX. Then the quotient X/Gis also a Hausdorff, second countable C∞-scheme. If Xis locally fair, or locally finitely presented, then so is X/G. Proof.Letx∈X, and write H=/braceleftbig γ∈G:γ(x) =x/bracerightbig . Then the G-orbitxG is|G|/|H|points. Since Xis Hausdorff and Gis finite, we can find an open neighbourhood RofxinXsuch thatRisH-invariant and R∩γ·R=∅for all γ∈G\H. AsXis aC∞-scheme, there is an open neighbourhood SofxinR with (S,OX|S) an affineC∞-scheme. Then T=/intersectiontext γ∈Hγ·Sis anH-invariant open neighbourhood of xinS. Choose an open neighbourhood UofxinTwith (U,OX|U) an affineC∞-scheme. DefineV=/intersectiontext γ∈Hγ·U. ThenVis anH-invariant open neighbourhood ofxinU⊆T⊆S⊆R⊆X. It is the intersection of the |H|affineC∞- subschemes ( γ·U,OX|γ·U) forγ∈Hinside the affine C∞-scheme (S,OX|S). 43Finite intersections of affine C∞-subschemes in an affine C∞-scheme are affine, as such intersections are fibre products and Spec : C∞Ringsop→LC∞RS preserves limits by Remark 4.21(b). Thus ( V,OX|V) is an affine C∞-scheme. SetW=/uniontext γH∈G/Hγ·V. ThenWis aG-invariant open neighbourhood ofxinX, and (W,OX|W) is the disjoint union of |G|/|H|affineC∞-schemes isomorphic to ( V,OX|V), so it is affine. We have shown that every x∈Xhas aG-invariant open neighbourhood W⊆XwithW= (W,OX|W) affine. Then W/Gis an open neighbourhood of xGinX/G. AsXis second countable, W is second countable and so Lindel¨ of. Thus W/Gis an affine C∞-scheme by Proposition 4.46. As we can cover X/Gby such open W/G, it is aC∞-scheme. IfXis locally fair, or locally finitely presented, we can do the argument above with S,U,V,W,W/Gfair, or finitely presented, using Proposition 2.22 forW/G, soX/Gis also locally fair, or locally finitely presented. 5 Modules over C∞-rings and C∞-schemes Nextwediscussmodulesover C∞-rings,andsheavesofmoduleson C∞-schemes. The author knows of no previous work on these, so all this section m ay be new, although much of it is a straightforward generalization of well known facts. 5.1 Modules over C∞-rings Definition 5.1. LetCbe aC∞-ring. A moduleMoverC, orC-module, is a module over Cregarded as a commutative R-algebra as in Definition 2.6, and morphisms of C-modules are morphisms of R-algebra modules. We will write µM:C×M→Mfor the multiplication map, and also write µM(c,m) =c·m forc∈Candm∈M. ThenC-modules form an abelian category, which we write as C-mod. The action of Con itself by multiplication makes Cinto aC-module, and moregenerally C⊗RVisaC-moduleforany R-vectorspace V. AC-moduleMis finitely generated ifitfitsintoanexactsequence C⊗Rn→M→0inC-mod, and finitely presented if it fits into an exact sequence C⊗Rm→C⊗Rn→M→0. BecauseC∞-rings such as C∞(Rn) are not noetherian, finitely generated C-modules generally need not be finitely presented. Now letφ:C→Dbe a morphism of C∞-rings. IfMis aC-module then φ∗(M) =M⊗CDis aD-module, and this induces a functor φ∗:C-mod→ D-mod. Also,any D-moduleNmayberegardedasa C-moduleφ∗(N) =Nwith C-actionµφ∗(N)(c,n) =µN(φ(c),n), and this defines a functor φ∗:D-mod→ C-mod. Note that φ∗:C-mod→D-mod takes finitely generated (or finitely presented) C-modules to finitely generated (or finitely presented) D-modules, butφ∗:D-mod→C-mod generally does not. Vector bundles Eover manifolds Xgive examples of modules over C∞(X). Example 5.2. LetXbe a manifold and E→Xbe a vector bundle, and write Γ∞(E) for the vector space of smooth sections eofE. This is a module over 44theC∞-ringC∞(X), multiplying functions on Xby sections of E. LetE,F→Xbe vector bundles over Xandλ:E→Fa morphism of vector bundles. Then λ∗: Γ∞(E)→Γ∞(F) defined by λ∗:e/ma√sto→λ◦eis a morphism of C∞(X)-modules. Now letX,Ybe manifolds and f:X→Ya (weakly) smooth map. Then f∗:C∞(Y)→C∞(X) is a morphism of C∞-rings. IfE→Yis a vector bundle over Y, thenf∗(E) is a vector bundle over X. Under the functor ( f∗)∗: C∞(Y)-mod→C∞(X)-mod of Definition 5.1, we see that ( f∗)∗/parenleftbig Γ∞(E)/parenrightbig = Γ∞(E)⊗C∞(Y)C∞(X) is isomorphic as a C∞(X)-module to Γ∞/parenleftbig f∗(E)/parenrightbig . IfE→Xis any vector bundle over a manifold Xthen by choosing sections e1,...,en∈Γ∞(E) forn≫0 such that e1|x,...,en|xspanE|xfor allx∈X we obtain a surjective morphism of vector bundles ψ:X×Rn→E, whose kernel is another vector bundle F. By choosing another surjective morphism φ:X×Rm→Fwe obtain an exact sequence of vector bundles X×Rmφ/d47/d47X×Rnψ/d47/d47E /d47/d470, which induces an exact sequence of C∞(X)-modules C∞(X)⊗RRmφ∗/d47/d47C∞(X)⊗RRnψ∗/d47/d47Γ∞(E) /d47/d470. Thus Γ∞(E) is a finitely presented C∞(X)-module. 5.2 Cotangent modules of C∞-rings Given aC∞-ringC, we will define the cotangent module ΩCofC. Although our definition of C-module only used the commutative R-algebra underlying the C∞-ringC, our definition of the particular C-module Ω Cdoes use the C∞-ring structure in a nontrivial way. It is a C∞-ring version of the module of relative differential forms orK¨ ahler differentials in Hartshorne [31, p. 172], and is an example of a construction for Fermat theories by Dubuc and Kock [ 25]. Definition 5.3. Suppose Cis aC∞-ring, andMaC-module. A C∞-derivation is anR-linear map d : C→Msuch that whenever f:Rn→Ris a smooth map andc1,...,cn∈C, we have dΦf(c1,...,cn) =n/summationtext i=1Φ∂f ∂xi(c1,...,cn)·dci. (5.1) Note that d is nota morphism of C-modules. We call such a pair M,d acotan- gent module forCif it has the universal property that for any C∞-derivation d′:C→M′, there exists a unique morphism of C-modulesλ:M→M′ with d′=λ◦d. There is a natural construction for a cotangent module: we take Mto be the quotient of the free C-module with basis of symbols d cforc∈C by theC-submodule spanned by all expressions of the form dΦ f(c1,...,cn)−/summationtextn i=1Φ∂f ∂xi(c1,...,cn)·dciforf:Rn→Rsmooth and c1,...,cn∈C. Thus 45cotangent modules exist, and are unique up to unique isomorphism. W hen we speak of ‘the’ cotangent module, we mean that constructed abov e. We write dC:C→ΩCfor the cotangent module of C. LetC,DbeC∞-rings with cotangent modules Ω C,dC, ΩD,dD, andφ:C→ Dbe a morphism of C∞-rings. Then we may regard Ω D=φ∗(ΩD) as aC- module, and d D◦φ:C→ΩDas aC∞-derivation. Thus by the universal property of Ω C, there exists a unique morphism of C-modules Ω φ: ΩC→ΩD with d D◦φ= Ωφ◦dC. This then induces a morphism of D-modules (Ω φ)∗: ΩC⊗CD→ΩD. Ifφ:C→D,ψ:D→Eare morphisms of C∞-rings then Ωψ◦φ= Ωψ◦Ωφ: ΩC→ΩE. Example 5.4. LetXbeamanifold. Thenthecotangentbundle T∗Xisavector bundle over X, so as in Example 5.2 it yields a C∞(X)-module Γ∞(T∗X). The exterior derivative d : C∞(X)→Γ∞(T∗X), d :c/ma√sto→dcis then aC∞-derivation, since equation (5.1) follows from d/parenleftbig f(c1,...,cn)/parenrightbig =/summationtextn i=1∂f ∂xi(c1,...,cn)dcn forf:Rn→Rsmooth and c1,...,cn∈C∞(X), which holds by the chain rule. It is easy to show that Γ∞(T∗X),d have the universal property in Definition 5.3, and so form a cotangent module for C∞(X). Now letX,Ybe manifolds, and f:X→Ya smooth map. Then f∗(T∗Y), T∗Xare vector bundles over X, and the derivative of fgives a vector bundle morphism d f:f∗(T∗Y)→T∗X. This induces a morphism of C∞(X)-modules (df)∗: Γ∞(f∗(T∗Y))→Γ∞(T∗X). This (df)∗is identified with (Ω f∗)∗under the natural isomorphism Γ∞(f∗(T∗Y))∼=Γ∞(T∗Y)⊗C∞(Y)C∞(X), where we identifyC∞(Y),C∞(X),f∗withC,D,φin Definition 5.3. The importance of Definition 5.3 is that it abstracts the notion of cot angent bundle of a manifold in a way that makes sense for any C∞-ring. Remark 5.5. There is a second way to define a cotangent-type module for a C∞-ringC, namely the module Kd CofK¨ ahler differentials of the underlying R-algebra of C. This is defined as for Ω C, but requiring (5.1) to hold only when f:Rn→Ris a polynomial. Since we impose many fewer relations, Kd Cis generally much larger than Ω C, so that Kd C∞(Rn)is not a finitely generated C∞(Rn)-module for n>0, for instance. Proposition 5.6. IfCis a finitely generated C∞-ring then ΩCis a finitely generated C-module. If Cis finitely presented, then ΩCis finitely presented. Proof.IfCis finitely generated we have an exact sequence 0 /d47/d47I /d47/d47C∞(Rn)φ/d47/d47C /d47/d470. (5.2) Writex1,...,xnfor the generators of C∞(Rn). Then any c∈Cmay be written asφ(f) for somef∈C∞(Rn), and (5.1) implies that dc= dΦf/parenleftbig φ(x1),...,φ(xn)/parenrightbig =/summationtextn i=1Φ∂f ∂xi(φ(x1),...,φ(xn))·d◦φ(xi). 46Hence the generators d cof ΩCforc∈CareC-linear combinations of d ◦φ(xi), i= 1,...,n, so ΩCis spanned by the d ◦φ(xi), and is finitely generated. Suppose Cis finitely presented. Then we have an exact sequence (5.2) with idealI= (f1,...,fm). We will define an exact sequence of C-modules C⊗RRmα/d47/d47C⊗RRnβ/d47/d47ΩC/d47/d470. (5.3) Write (a1,...,am), (b1,...,bn) for bases of Rm,Rn. AsC⊗RRm,C⊗RRnare freeC-modules, the C-module morphisms α,βare specified uniquely by giving α(ai) fori= 1,...,mandβ(bj) forj= 1,...,n, which we define to be α:ai/ma√sto−→/summationtextn j=1Φ∂fi ∂xj/parenleftbig φ(x1),...,φ(xn)/parenrightbig ·bjandβ:bj/ma√sto−→dC/parenleftbig φ(xj)/parenrightbig . Then fori= 1,...,mwe have β◦α(ai) =/summationtextn j=1Φ∂fi ∂xj/parenleftbig φ(x1),...,φ(xn)/parenrightbig ·dC/parenleftbig φ(xj)/parenrightbig = dC/parenleftbig Φfi/parenleftbig φ(x1),...,φ(xn)/parenrightbig/parenrightbig = dC◦φ/parenleftbig Φfi(x1,...,xn)/parenrightbig = dC◦φ/parenleftbig fi(x1,...,xn)) = d C(0) = 0, using (5.1) in the second step, φa morphism of C∞-rings in the third, the definition of C∞(Rn) asaC∞-ringin the fourth, and fi(x1,...,xn)∈I= Kerφ in the fifth. Hence β◦α= 0, and (5.3) is a complex. Thusβinducesβ∗: (C⊗RRn)/α(C⊗RRm)→ΩC. We will show β∗is an isomorphism, so that (5.3) is exact. Define d : C→(C⊗RRn)/α(C⊗RRm) by d/parenleftbig φ(h)/parenrightbig =/summationtextn j=1Φ∂h ∂xj/parenleftbig φ(x1),...,φ(xn)/parenrightbig ·bj+α(C⊗RRm).(5.4) Hereeveryc∈Cmaybe written as φ(h) forsomeh∈C∞(Rn) asφis surjective. To show (5.4) is well-defined we must show the right hand side is indepen dent of the choice of hwithφ(h) =c, that is, we must show that the right hand side is zero ifh∈I. It is enough to check this when his a generator f1,...,fmof I, and this holds by definition of α. Hence d in (5.4) is well-defined. It is easy to see that d is a C∞-derivation, and that β∗◦d = d C. So by the universal property of Ω C, there is a unique C-module morphism ψ: ΩC→ (C⊗RRn)/α(C⊗RRm)withd =ψ◦dC. Thusβ∗◦ψ◦dC=β∗◦d = dC= idΩC◦dC, so as Imd Cgenerates Ω Cas anC-module we see that β∗◦ψ= idΩC. Similarly ψ◦β∗is the identity, so ψ,β∗are inverse, and β∗is an isomorphism. Therefore (5.3) is exact, and Ω Cis finitely presented. Cotangent modules behave well under localization. Proposition 5.7. LetCbe aC∞-ring,S⊆C,andD=C[s−1:s∈S]be the localization of CatSwith projection π:C→D,as in Definition 2.13. Then (Ωπ)∗: ΩC⊗CD→ΩDis an isomorphism of D-modules. 47Proof.Let ΩC,ΩDbe constructed as in Definition 5.3. As D=C[s−1:s∈S] isCtogether with an extra generator s−1and an extra relation s·s−1= 1 for eachs∈S, we see that the D-module Ω Dmay be constructed from Ω C⊗CD by adding an extra generator d( s−1) and an extra relation d( s·s−1−1) = 0 for eachs∈S. But using (5.1) and s·s−1= 1 inD, we see that this extra relation is equivalent to d( s−1) =−(s−1)2ds. Thus the extra relations exactly cancel the effect of adding the extra generators, so (Ω π)∗is an isomorphism. Here is a useful exactness property of cotangent modules. Theorem 5.8. Suppose we are given a pushout diagram of C∞-rings: Cβ/d47/d47 α/d15/d15E δ/d15/d15 Dγ/d47/d47F,(5.5) so thatF=D∐CE. Then the following sequence of F-modules is exact: ΩC⊗C,γ◦αF(Ωα)∗⊕−(Ωβ)∗/d47/d47ΩD⊗D,γF⊕ ΩE⊗E,δF(Ωγ)∗⊕(Ωδ)∗/d47/d47ΩF/d47/d470.(5.6) Here(Ωα)∗: ΩC⊗C,γ◦αF→ΩD⊗D,γFis induced by Ωα: ΩC→ΩD,and so on. Note the sign of −(Ωβ)∗in(5.6). Proof.By Ωψ◦φ= Ωψ◦Ωφin Definition 5.3 and commutativity of (5.5) we have Ωγ◦Ωα= Ωγ◦α= Ωδ◦β= Ωδ◦Ωβ: ΩC→ΩF. Tensoring with Fthen gives (Ωγ)∗◦(Ωα)∗= (Ωδ)∗◦(Ωβ)∗: ΩC⊗CF→ΩF. Asthe compositionofmorphisms in (5.6) is (Ω γ)∗◦(Ωα)∗−(Ωδ)∗◦(Ωβ)∗, this implies (5.6) is a complex. For simplicity, first suppose C,D,E,Fare finitely presented. Use the nota- tion of Example 2.23 and the proof of Proposition 2.24, with exact seq uences (2.3) and (2.4), where I= (h1,...,hi)⊂C∞(Rl),J= (d1,...,dj)⊂C∞(Rm) andK= (e1,...,ek)⊂C∞(Rn). ThenLis given by (2.5). Applying the proof of Proposition 5.6 to (2.3)–(2.4) yields exact sequences of F-modules F⊗RRiǫ1/d47/d47F⊗RRlζ1/d47/d47ΩC⊗CF /d47/d470,(5.7) F⊗RRjǫ2/d47/d47F⊗RRmζ2/d47/d47ΩD⊗DF /d47/d470,(5.8) F⊗RRkǫ3/d47/d47F⊗RRnζ3/d47/d47ΩE⊗EF /d47/d470,(5.9) F⊗RRj+k+lǫ4/d47/d47F⊗RRm+n=F⊗RRm⊕F⊗RRnζ4/d47/d47ΩF/d47/d470,(5.10) where for (5.7)–(5.9) we have tensored (5.3) for C,D,EwithF. DefineF-module morphisms θ1:F⊗RRl→F⊗RRm,θ2:F⊗RRl→F⊗RRn byθ1(a1,...,al) = (b1,...,bm),θ2(a1,...,al) = (c1,...,cn) with bq=l/summationdisplay p=1Φ∂fp ∂yq(ξ(y1),...,ξ(ym))·ap, cr=l/summationdisplay p=1Φ∂gp ∂yr(ξ(z1),...,ξ(zn))·ap, 48forap,bq,cr∈F. Now consider the diagram F⊗RRj⊕ F⊗RRk⊕ F⊗RRl ǫ 4=/parenleftBigǫ20θ1 0ǫ3−θ2/parenrightBig/d47/d47 (0 0ζ1) /d15/d15F⊗RRm⊕ F⊗RRnζ4/d47/d47 /parenleftBigζ20 0ζ3/parenrightBig /d15/d15ΩF/d47/d47 idΩF0 ΩC⊗CF/parenleftbigg (Ωα)∗ −(Ωβ)∗/parenrightbigg /d47/d47ΩD⊗DF⊕ ΩE⊗EF((Ωγ)∗(Ωδ)∗)/d47/d47ΩF/d47/d470,(5.11) using matrix notation. The top line is the exact sequence (5.10), whe re the sign in−θ2comes fromthe sign of gpin the generators fp(y1,...,ym)−gp(z1,...,zn) ofLin (2.5). The bottom line is the complex (5.6). The left hand square commutes as ζ2◦ǫ2=ζ3◦ǫ3= 0 by exactness of (5.8)– (5.9)andζ2◦θ1= (Ωα)∗◦ζ1followsfrom α◦φ(xp) =ψ(fp), andζ3◦θ2= (Ωβ)∗◦ζ1 follows from β◦φ(xp) =χ(gp). The right hand square commutes as ζ4and (Ωγ)∗◦ζ2act onF⊗RRmby (a1,...,am)/ma√sto→/summationtextm q=1aqdF◦ξ(yq), andζ4and (Ωδ)∗◦ζ3act onF⊗RRnby (b1,...,bn)/ma√sto→/summationtextn r=1brdF◦ξ(zr). Hence (5.11) is commutative. The columns are surjective since ζ1,ζ2,ζ3are surjective as (5.7)–(5.9) are exact and identities are surjective. The bottom right morphism/parenleftbig (Ωγ)∗(Ωδ)∗/parenrightbig in (5.11) is surjective as ζ4is and the right hand square commutes. Also surjectivity of the middle column implies that it maps Ker ζ4surjectively onto Ker/parenleftbig (Ωγ)∗(Ωδ)∗/parenrightbig . But Kerζ4= Imǫ4as the top row is exact, so as the left hand square commutes we see that/parenleftbig (Ωα)∗−(Ωβ)∗/parenrightbigTsurjects onto Ker/parenleftbig (Ωγ)∗(Ωδ)∗/parenrightbig , and the bottom row of (5.11) is exact. This proves the theorem for C,D,E,Ffinitely presented. For the general case we can use the same proof, but allowing i,j,k,l,m,n infinite. Here is an example of the situation of Theorem 5.8 for manifolds. Example 5.9. LetW,X,Y,Z,e,f,g,h be as in Theorem 3.5, so that (3.1) is a Cartesian square of manifolds and (3.2) a pushout square of C∞-rings. We have the following sequence of morphisms of vector bundles on W: 0 /d47/d47(g◦e)∗(T∗Z)e∗(dg∗)⊕−f∗(dh∗)/d47/d47e∗(T∗X)⊕f∗(T∗Y)de∗⊕df∗ /d47/d47T∗W /d47/d470.(5.12) Here dg:TX→g∗(TZ) is a morphism of vector bundles over X, and dg∗: g∗(T∗Z)→T∗Xis the dual morphism, and e∗(dg∗) : (g◦e)∗(T∗Z)→e∗(T∗X) is the pullback of this dual morphism to W. Sinceg◦e=h◦f, we have de∗◦e∗(dg∗) = df∗◦f∗(dh∗), and so (5.12) is a complex. As g,haretransverseand(3.1)isCartesian,(5.12)isexact. Sopassing to smooth sections in (5.12) we get an exact sequence of C∞(W)-modules: 0 /d47/d47Γ∞/parenleftbig (g◦e)∗(T∗Z)/parenrightbig(e∗(dg∗)⊕ −f∗(dh∗))∗/d47/d47Γ∞/parenleftbig e∗(T∗X) ⊕f∗(T∗Y)/parenrightbig(de∗⊕ df∗)∗/d47/d47Γ∞(T∗W) /d47/d470. The final four terms are the exact sequence (5.6) for the pushou t diagram (3.2). 495.3 Sheaves of OX-modules on a C∞-ringed space (X,OX) We define sheaves of OX-modules on a C∞-ringed space, following [31, §II.5]. Definition 5.10. Let (X,OX) be aC∞-ringed space. A sheaf ofOX-modules , or simply anOX-module,EonXassigns a module E(U) over the C∞-ring OX(U) for each open set U⊆X, and a linear map EUV:E(U)→E(V) for each inclusion of open sets V⊆U⊆X, such that the following commutes OX(U)×E(U) ρUV×EUV/d15/d15µE(U)/d47/d47E(U) EUV/d15/d15 OX(V)×E(V)µE(V)/d47/d47E(V),(5.13) and all this data E(U),EUVsatisfies the sheaf axioms in Definition 4.1. Amorphism of sheaves of OX-modulesφ:E→Fassigns a morphism of OX(U)-modulesφ(U) :E(U)→F(U) for each open set U⊆X, such that φ(V)◦EUV=FUV◦φ(U) for each inclusion of open sets V⊆U⊆X. Then OX-modules form an abelian category, which we write as OX-mod. AnOX-moduleEis called a vector bundle of rank nif we may cover Xby openU⊆XwithE|U∼=OX|U⊗RRn. In Definition 4.7 we defined finesheavesEon a topological space X. In§4.7 we gave sufficient conditions for when a C∞-ringed space X= (X,OX) hasOX fine, which hold if Xis an affine C∞-scheme with XLindel¨ of. Now if OXis fine, then anyOX-moduleEis also fine, since partitions of unity in OXinduce partitions of unity in Hom(E,E). As in Voisin [69, Prop.4.36], a fundamental propertyof fine sheaves Eis that their cohomology groups Hi(E) are zero for all i >0. This means that H0is an exact functor on fine sheaves, rather than just left exact, s inceH1measures the failure of H0to be right exact. If Xis second countable then ( U,OX|U) is a Lindel¨ of affine C∞-scheme for all open U⊆X. Thus we deduce: Proposition 5.11. Let(X,OX)be an affine C∞-scheme with XLindel¨ of, and ··· /d47/d47Eiφi/d47/d47Ei+1φi+1/d47/d47Ei+2 /d47/d47··· be an exact sequence in OX-mod. Then ··· /d47/d47Ei(X)φi(X)/d47/d47Ei+1(X)φi+1(X)/d47/d47Ei+2(X) /d47/d47··· is an exact sequence of OX(X)-modules. If Xis also second countable then the following is an exact sequence of OX(U)-modules for all open U⊆X: ··· /d47/d47Ei(U)φi(U)/d47/d47Ei+1(U)φi+1(U)/d47/d47Ei+2(U) /d47/d47···. Remark 5.12. Recall that a C∞-ringChas an underlying commutative R- algebra, and a module over Cis a module over this R-algebra, by Definitions 2.6 and 5.1. Thus, by truncating the C∞-ringsOX(U) to commutative R-algebras, 50regarded as rings, a C∞-ringed space ( X,OX) has an underlying ringed space in the usual sense of algebraic geometry [31, p. 72], [30, §0.4]. Our definition ofOX-modules are simply OX-modules on this underlying ringed space [31, §II.5], [30,§0.4.1]. Thus we can apply results from algebraic geometry without change, for instance that OX-mod is an abelian category, as in [31, p. 202]. Definition 5.13. Letf= (f,f♯) : (X,OX)→(Y,OY) be a morphism of C∞-ringed spaces, and Ebe anOY-module. Define the pullbackf∗(E) by f∗(E) =f−1(E)⊗f−1(OY)OX, wheref−1(E) is as in Definition 4.5, a sheaf of modules over the sheaf of C∞-ringsf−1(OY) onX, and the tensor product uses the morphism f♯:f−1(OY)→OX. Ifφ:E→Fis a morphism of OY-modules we have a morphism of OX-modulesf∗(φ) =f−1(φ)⊗idOX:f∗(E)→f∗(F). Remark 5.14. Pullbacksf∗(E) are a kind of fibre product, and may be char- acterized by a universal property in OX-mod. So they should be regarded as beingunique up to canonical isomorphism , rather than unique. One can give an explicit construction for pullbacks, or use the Axiom of Choice to c hoose f∗(E) for allf,E, and so speak of ‘the’ pullback f∗(E). However, it may not be possible to make these choices strictly functorial in f. That is, if f:X→Y,g:Y→Zare morphisms and E∈OZ-mod then (g◦f)∗(E),f∗(g∗(E)) are canonically isomorphic in OX-mod, but may not be equal. We will write If,g(E) : (g◦f)∗(E)→f∗(g∗(E)) for these canonical isomorphisms, as in Remark 4.6(b). Then If,g: (g◦f)∗⇒f∗◦g∗is a natural isomorphism of functors. It is common to ignore this point and identif y (g◦f)∗ withf∗◦g∗. Vistoli [68] makes careful use of natural isomorphisms ( g◦f)∗⇒ f∗◦g∗in his treatment of descent theory. Whenfis the identity id X:X→XandE∈OX-mod we do not require id∗ X(E) =E, but asEis a possible pullback for id∗ X(E) there is a canonical isomorphism δX(E) : id∗ X(E)→E, and then δX: id∗ X⇒idOX-modis a natural isomorphism of functors. By Grothendieck [30, §0.4.3.1] we have: Proposition 5.15. LetX,YbeC∞-ringed spaces and f:X→Ya morphism. Then pullback f∗:OY-mod→OX-modis aright exact functor between abelian categories. That is, if Eφ−→Fψ−→G → 0is exact inOY-modthen f∗(E)f∗(φ)−→f∗(F)f∗(ψ)−→f∗(G)→0is exact inOX-mod. In generalf∗is not exact, or left exact, unless f:X→Yis flat. 5.4 Sheaves on affine C∞-schemes, MSpecandΓ In§4.4 we defined Spec : C∞Ringsop→LC∞RS. In a similar way, if Cis a C∞-ring and (X,OX) = Spec Cwe can define MSpec : C-mod→OX-mod, a spectrum functor for modules. Definition 5.16. Let (X,OX) = Spec Cfor someC∞-ringCandMbe aC- module. We will define an OX-moduleE= MSpecM. For each open U⊆X, 51defineE(U) to be the R-vector space of functions e:U→/coproducttext x∈U(M⊗CCx) with e(x)∈M⊗CCxfor allx∈U, and such that Umay be covered by open sets W⊆U⊆Xfor which there exist m∈Mwithe(x) =m⊗1∈M⊗CCxfor all x∈W. Here the Cx-moduleM⊗CCxis defined using the C-module structure onMand the projection πx:C→Cx. Definition 4.16 defines OX(U) as a set of functions U→/coproducttext x∈UCx. Define anOX(U)-module structure µE(U):OX(U)×E(U)→E(U) onE(U) by µE(U)(s,e) :x/ma√sto−→s(x)·e(x), for alls∈ OX(U),e∈ E(U) andx∈U. For open V⊆U⊆X, define EUV:E(U)→E(V) byEUV:e/ma√sto→e|V. It is now easy to check that Eis a sheaf ofOX-modules on X. Define MSpec M=EinOX-mod. An equivalent way to define MSpec Mis as the sheafification of the presheaf U/ma√sto→M⊗COX(U). The definition above performs the sheafification explicitly. Now letα:M→Nbe a morphism in C-mod, and setE= MSpecMand F= MSpecN. For each open U⊆X, defineλ(U) :E(U)→F(U) by λ(U)(e) :x/ma√sto→(α⊗id)(e(x)) forx∈U, whereα⊗id mapsM⊗CCx→N⊗CCx. It is easy to check that λ(U) is an OX(U)-module morphism and λ(V)◦EUV=FUV◦λ(U) :E(U)→F(V) for all openV⊆U⊆X. Henceλ:E→Fis a morphism in OX-mod. Define MSpecα=λ, so that MSpec α: MSpecM→MSpecN. This defines a functor MSpec : C-mod→OX-mod. It is an exact functor of abelian categories, since M/ma√sto→M⊗CCxis an exact functor C-mod→Cx-mod for each x∈X, as the localization πx:C→Cxis a flat morphism of R-algebras. Definition 5.17. LetCbe aC∞-ring, and ( X,OX) = Spec C. IfEis anOX- module thenE(X) is a module over OX(X), so using Ψ C:C→Γ(SpecC) = OX(X) we may regard E(X) as aC-module. Define Γ( E) to be the C-module E(X). Ifα:E → F is a morphism of OX-modules then Γ( α) :=α(X) : E(X)→F(X) is a morphism Γ( α) : Γ(E)→Γ(F) inC-mod. This defines the global sections functor Γ :OX-mod→C-mod. In general Γ is a left exact functor of abelian categories, but may n ot be right exact. However, if Xis Lindel¨ of (for example, if Cis finitely or countably generated) then Proposition 5.11 shows that Γ is an exact functor . Now Γ◦MSpec is a functor C-mod→C-mod. For each C-moduleMand m∈M, define Ψ M(m) :X→/coproducttext x∈XM⊗CCxby ΨM(m) :x/ma√sto→m⊗1Cx∈ M⊗CCx. Then Ψ M(m)∈MSpecM(X) = Γ◦MSpecMby Definition 5.16, so ΨM:M→Γ◦MSpecMis a linear map, and in fact a C-module morphism. Itisfunctorialin M,sothattheΨ MforallMdefineanaturaltransformation Ψ : idC-mod⇒Γ◦MSpec of functors id C-mod,Γ◦MSpec : C-mod→C-mod. Here are the analogues of Lemma 4.18 and Theorem 4.20: Lemma 5.18. In Definition 5.16,the stalk (MSpecM)x=ExofMSpecMat x∈Xis naturally isomorphic to M⊗CCx,as modules over Cx∼=OX,x. 52Proof.Elements ofExare∼-equivalence classes [ U,e] of pairs (U,e), whereU is an open neighbourhood of xinXande∈E(U), and (U,e)∼(U′,e′) if there exists open x∈V⊆U∩U′withe|V=e′|V. Define a Cx-module morphism Π :Ex→M⊗CCxby Π : [U,e]/ma√sto→e(x). Proposition 2.14 shows that Cx∼=C/IforIthe ideal in (2.2). Hence M⊗C Cx∼=M/(I·M), and thus every element of M⊗CCxis of the form m⊗1Cx for somem∈M. But ΨM(m)∈E(X), so that [ X,ΨM(m)]∈Ex, with Π : [X,ΨM(m)]/ma√sto→m⊗1Cx. Hence Π :Ex→M⊗CCxis surjective. Suppose [U,e]∈Exwith Π([U,e]) = 0∈M⊗CCx. Ase∈E(U), there exist openx∈V⊆Uandm∈Mwithe(x′) =m⊗1Cx′∈M⊗CCx′for allx′∈V. Thenm⊗1Cx=e(x) = Π([U,e]) = 0 inM⊗CCx, som∈I·M⊆M, and we may writem=/summationtextk a=1ia·maforia∈Iandma∈M. By (2.2) we may choose d1,...,dk∈Cwithx(da)/\e}atio\slash= 0 andia·da= 0 inCfora= 1,...,k. SetW={x′∈V:x′(da)/\e}atio\slash= 0, a= 1,...,k}, so thatWis an open neighbourhood of xinU. Ifx′∈Wthenx′(da)/\e}atio\slash= 0, soπx′(da) is invertible in Cx′. Butia·da= 0, soπx′(ia) = 0 inCx′fora= 1,...,k. Asm=/summationtextk a=1ia·ma it follows that e(x′) =m⊗1Cx′= 0 inM⊗CCx′for allx′∈W. Thuse|W= 0 in E(W), so [U,e] = [W,e|W] = 0 inEx. Therefore Π :Ex→M⊗CCxis injective, and so an isomorphism. Theorem 5.19. LetCbe aC∞-ring, and (X,OX) = Spec C. Then Γ : OX-mod→C-modisright adjoint toMSpec : C-mod→OX-mod. That is, for allM∈C-modandE∈OX-modthere are inverse bijections HomC-mod(M,Γ(E))LM,E/d47/d47HomOX-mod(MSpecM,E), RM,E/d111/d111 (5.14) which are functorial in M,E. WhenE= MSpecMwe have ΨM=RM,E(idE), so thatΨMis the unit of the adjunction between ΓandMSpec. Proof.LetM∈C-mod andE∈OX-mod, and setD= MSpecM. DefineRM,E in (5.14) by, for each morphism α:D→EinOX-mod, taking RM,E(α) :M→ Γ(E) to be the composition MΨM/d47/d47Γ◦MSpecM= Γ(D)Γ(α)/d47/d47Γ(E). For the last part, if E= MSpecMthen ΨM=RM,E(idE) as Γ(id E) = idΓ(E). Letβ:M→Γ(E) be a morphism in C-mod. We will construct a morphism λ:D→EinOX-mod, and set LM,E(β) =λ. Letx∈X. Consider the diagram M⊗CC=M id⊗πx/d15/d15β/d47/d47Γ(E) σx/d15/d15 M⊗CCx∼=Dxβx/d47/d47Ex(5.15) inC-mod, where the isomorphism M⊗CCx∼=Dxcomes from Lemma 5.18. HereExis the stalk ofEatx, andσx: Γ(E) =E(X)→Extakes stalks at 53x. TheC-action on Γ(E) factors via CΨC−→OX(X), and the C-action onEx factors via CΨC−→OX(X)π−→OX,x, andβ,σxare both C-module morphisms. ButOX,x∼=Cxby Lemma 4.18, so σx◦β:M→Exis aC-module morphism, where the C-action onExfactors via Cπx−→Cx. Hence there is a unique OX,x- module morphism βx:Dx→Exmaking (5.15) commute. For each open U⊆X, defineλ(U) :D(U)→E(U) byλ(U)d:x/ma√sto→βx(d(x)) ford∈D(U) andx∈U⊆X, andd(x)∈Dx, andβx(d(x))∈Ex. Here asE is a sheaf we may identify elements of E(U) with maps e:U→/coproducttext x∈UExwith e(x)∈Exforx∈U, such that esatisfies certain local conditions in U. Ifd∈D(U) = MSpec M(U) andx∈Uthen by Definition 5.16 we may coverUby openW⊆Ufor which there exist m∈Mwithd(x) =m⊗1Cxin M⊗CCxfor allx∈W. Therefore λ(U)dmapsx/ma√sto→σx(β(m)) for allx∈Wby (5.15), soλ(U)dis a section β(m)|WofEonW. Henceλ(U)dis a section of E|U, as suchWcoverU, andλ(U) :D(U)→E(U) is well defined. Asβxis anOX,x-module morphism for all x∈U,λ(U) :D(U)→E(U) is anOX(U)-module morphism. The definition of λ(U) is clearly compatible with restriction to open V⊆U⊆X. Thus the λ(U) for all open U⊆Xdefine a sheaf morphism λ:D→EinOX-mod. SetLM,E(β) =λ. This defines LM,Ein (5.14). A very similar proof to that of Theorem 4.20 shows that LM,E,RM,Eare inverse maps, so they are bijections, and that they are functoria l inM,E. We show that Γ is a right inverse for MSpec: Proposition 5.20. LetCbe aC∞-ring, and (X,OX) = Spec C,andEbe anOX-module. Set M= Γ(E)inC-mod,and write ΨE=LM,E(idM). Then ΨE: MSpec◦Γ(E)→Eis an isomorphism in OX-mod,for anyE. These isomorphisms ΨEare functorial inE,and so define a natural isomor- phismΨ : MSpec◦Γ⇒idOX-modof functorsOX-mod→OX-mod. Proof.SetD= MSpecM= MSpec◦Γ(E), and letx∈X. Then by definition of ΨE=LM,E(idM) :D→Ein the proof of Theorem 5.19, as in (5.15) the stalk map Ψ E,x:Dx→Exis the unique morphism of modules over Cx∼=OX,x making the following diagram of C-modules commute: M⊗CC=M id⊗πx/d15/d15idM/d47/d47M= Γ(E) σx/d15/d15 M⊗CCx∼=DxΨE,x/d47/d47Ex.(5.16) Let [U,e]∈Ex, so thatx∈U⊆Xis open and e∈E(U). By Definition 4.13 there exists c∈Csuch thatx(c)/\e}atio\slash= 0 andy(c) = 0 for all y∈X\U. Choose smooth f:R→Rsuch thatf= 0 near 0 in Randf= 1 nearx(c) inR. Setc′= Φf(c), where Φ f:C→Cis theC∞-ring operation. Then η= ΨC(c′)∈OX(X), and there exist open neighbourhoods VofX\UandW ofxinXwithη|V= 0 andη|W= 1. Clearly V∩W=∅, sox∈W⊆U. We haveη|U·e∈E(U), with (η|U·e)|U∩V= 0 and (η|U·e)|W=e|W. 54Since{U,V}is an open cover of Xand (η|U·e)|U∩V= 0 = 0|U∩V, by the sheaf property of Ethere is a unique e′∈E(X) withe′|U=η|U·eande′|V= 0. Thene′|W= (η|U·e)|W=e|W. Thus σx(e′) = [X,e′] = [W,e′|W] = [W,e|W] = [U,e] inEx. Henceσx: Γ(E)→Exis surjective, so Ψ E,x:Dx→Exis surjective by (5.16), asπx:C→Cxis surjective by Proposition 2.14. Supposed∈Dxwith Ψ E,x(d) = 0. We may write m⊗1Cx∼=dunder the isomorphism M⊗CCx∼=Dxfor somem∈M, and then (5.16) gives σx(m) = ΨE,x(d) = 0. Hence there exists open x∈U⊆Xwithm|U= 0. As above we may construct η∈OX(X) and open V,W⊆XwithX\U⊆V, x∈W⊆U,η|V= 0 andη|W= 1. Then η·m= 0 inMasm|U= 0,η|V= 0 withU∪V=X, andπx(η) = 1CxinCxasη= 1 nearxinX. Hence m⊗1Cx=1Cx·(m⊗1Cx)=πx(η)·(m⊗1Cx)=(η·m)⊗1Cx=0⊗1Cx=0 inM⊗CCx. Therefore d= 0 inDx, and Ψ E,x:Dx→Exis injective, and so an isomorphism. As this holds for all x∈X, ΨE:D→Eis an isomorphism, proving the first part of the proposition. The second part follows f romLM,E functorial in M,Ein Theorem 5.19. As for quasicoherent sheaves in conventional algebraic geometry , we define: Definition 5.21. LetX= (X,OX) be aC∞-scheme, andEbe anOX-module. We callEquasicoherent if we may cover Xwith open U⊆Xsuch that (U,OX|U)∼=SpecCandE|U∼=MSpecMfor someC∞-ringCandC-moduleM. We write qcoh( X) for the category of quasicoherent sheaves on X. If (X,OX) is aC∞-scheme andEanOX-module, we can cover Xby open U⊆Xwith (U,OX|U)∼=SpecCaffine, and then Proposition 5.20 shows that E|U∼=MSpecMforM=E(U). Thus we have: Corollary 5.22. LetX= (X,OX)be aC∞-scheme. Then every OX-module Eis quasicoherent, so that qcoh(X) =OX-mod. Remark 5.23. (a) In conventional algebraic geometry, as in Hartshorne [31, §II.5], ifRis a ring and ( X,OX) = SpecRthe corresponding affine scheme, we also have functors MSpec : R-mod→OX-mod and Γ :OX-mod→R-mod. In C∞-algebraic geometry, as in Proposition 5.20, Γ is a right inverse for MS pec, but may not be a left inverse. But in algebraic geometry the opposite happens, as Γ is a left inverse for MSpec [31, Cor. II.5.5], but may not be a right in verse. The fact that Γ is a right inverseforMSpec in C∞-algebraicgeometrymeans that allOX-modules on a C∞-scheme (X,OX) are quasicoherent, so quasico- herence is not a very useful idea. But in algebraic geometry, as Γ is n ot a right inverse for MSpec, this is false: there are many examples of schemes ( X,OX) andOX-modulesEwhich are not quasicoherent. For instance, we may take X=A1andE(U) = 0 if 0∈U,E(U) =OX(U) if 0/∈Ufor all open U⊆X. 55In§5.5 we will define a module Mover aC∞ringCto becomplete if M∼=Γ◦MSpecM. Then Γ is a left inverse for MSpec on the subcategory C-modco⊂C-mod of complete C-modules. In general C-modules need not be complete. But in conventional algebraic geometry, as Γ is a left inver se for MSpec allR-modules are complete, so completeness is not a useful idea. (b)Inconventionalalgebraicgeometryonedefines coherent sheaves [31,§II.5]to be quasicoherent sheaves Elocally modelled on MSpec MforMa finitely gener- atedC-module. However, coherent sheaves are only well behaved on noetherian schemes, and most interesting C∞-rings, such as C∞(Rn) forn >0, are not noetherian R-algebras. Because of this, coherent sheaves do not seem to be a useful idea in C∞-algebraic geometry (for instance, coh( X) is not closed under kernelsin qcoh( X), and is not an abelian category), and we do not discussthem. We can understand the pullback functor f∗in Definition 5.13 explicitly in terms of modules over the corresponding C∞-rings: Proposition 5.24. LetC,DbeC∞-rings,φ:D→Ca morphism, M,NbeD- modules, and α:M→Na morphism of D-modules. Write X= SpecC, Y= SpecD, f= Specφ:X→Y,andE= MSpecM,F= MSpecNinqcoh(Y). Then there are natural isomorphisms f∗(E)∼=MSpec(M⊗DC)andf∗(F)∼= MSpec(N⊗DC)inqcoh(X). These identify MSpec(α⊗idC) : MSpec(M⊗D C)→MSpec(N⊗DC)withf∗(MSpecα) :f∗(E)→f∗(F). Proof.WriteX= (X,OX),Y= (Y,OY) andf= (f,f♯). ThenEis the sheafificationofthepresheaf V/ma√sto→M⊗DOY(V), andf−1(E) isthe sheafification of the presheaf U/ma√sto→limV⊇f(U)E(V), andf−1(OY) is the sheafification of the presheafU/ma√sto→limV⊇f(U)OY(V). Inf∗(E) =f−1(E)⊗f−1(OY)OX, these three sheafifications combine into one, so f∗(E) is the sheafification of the presheaf U/ma√sto→limV⊇f(U)(M⊗DOY(V))⊗OY(V)OX(U). But (M⊗DOY(V))⊗OY(V)OX(U)∼=M⊗DOX(U)∼=(M⊗DC)⊗COX(U), sothis iscanonicallyisomorphictothepresheaf U/ma√sto→(M⊗DC)⊗COX(U) whose sheafification is MSpec( M⊗DC). This gives a natural isomorphism f∗(E)∼= MSpec(M⊗DC). The same holds for N. The identification of MSpec( α⊗idC) andf∗(MSpecα)followsbypassingfrommorphismsofpresheavestomorphisms of the associated sheaves. 5.5 Complete modules over C∞-rings Here are the module analogues of Definition 4.35 and Theorem 4.36(b) ,(c). Definition 5.25. LetCbe aC∞-ring, andMaC-module. We call Mcomplete if ΨM:M→Γ◦MSpecMin Definition 5.17 is an isomorphism. WriteC-modcofor the full subcategory of complete C-modules in C-mod. IfMis aC-module then applying Γ to Proposition 5.20 shows that Γ(ΨMSpecM) : Γ◦MSpec(Γ◦MSpecM)−→Γ◦MSpecM 56is an isomorphism. From the definitions we can show that Ψ Γ◦MSpecM= Γ(ΨMSpecM)−1. Thus Γ◦MSpecMis complete, for any C-moduleM. De- fine a functor Rco all= Γ◦MSpec : C-mod→C-modco. Theorem 5.26. LetCbe aC∞-ring, andX= (X,OX) = Spec C. Then (a)MSpec|C-modco:C-modco→qcoh(X)is an equivalence of categories. (b)Rco all:C-mod→C-modcois left adjoint to the inclusion functor inc : C-modco֒→C-mod. That is,Rco allis areflection functor . Proof.For (a), ifM,Nare complete C-modules then putting E= MSpecNin Theorem 5.19 and using Γ ◦MSpecN∼=N, equation (5.14) shows that MSpec =LM,E: Hom C-modco(M,N)−→HomOX-mod(MSpecM,MSpecN) isabijection, wherethedefinitionof LM,EagreeswiththedefinitionofMSpecon morphisms in this case. Thus MSpec is full and faithful on complete C-modules. IfE ∈OX-mod = qcoh( X) thenE∼=MSpec◦Γ(E) by Proposition 5.20. Thus Γ(E)∼=Γ◦MSpec◦Γ(E), so Γ(E) is complete by Definition 5.25. Hence E∼=MSpec|C-modco[Γ(E)], and the essential image of MSpec |C-modcois qcoh(X). Therefore MSpec |C-modcois an equivalence of categories. For (b), let M,NbeC-modules with Ncomplete. Then we have bijections HomC-modco/parenleftbig Rco all(M),N/parenrightbig∼=HomC-mod/parenleftbig Γ◦MSpecM,Γ◦MSpecN/parenrightbig ∼=HomOX-mod/parenleftbig MSpec◦Γ◦MSpecM,MSpecN/parenrightbig ∼=HomOX-mod/parenleftbig MSpecM,MSpecN/parenrightbig (5.17) ∼=HomC-mod/parenleftbig M,Γ◦MSpecN/parenrightbig∼=HomC-mod/parenleftbig M,N/parenrightbig =Hom C-mod/parenleftbig M,inc(N)/parenrightbig , usingN∼=Γ◦MSpecNasNis complete in the first and fifth steps, Theorem 5.19 in the second and fourth, and Proposition 5.20 in the third. The b ijections (5.17) arefunctorial in M,Naseach step is. Hence Rco allis left adjoint to inc. Proposition 5.27. LetCbe aC∞-ring and (X,OX) = Spec C,and suppose Xis Lindel¨ of. Then C-modcois closed under kernels, cokernels and extensions inC-mod,that is,C-modcois an abelian subcategory of C-mod. Proof.As in§5.4, MSpec : C-mod→OX-mod is an exact functor, and as Xis Lindel¨ of Γ :OX-mod→C-mod is also exact by Proposition 5.11. Hence Rco all= Γ◦MSpec : C-mod→C-mod is an exact functor. Let 0 →M1→M2→M3be exact in C-mod withM2,M3complete. Then we have a commutative diagram 0 /d47/d47M1 ΨM1/d15/d15/d47/d47M2 ΨM2∼=/d15/d15/d47/d47M3 ΨM3∼=/d15/d15 0 /d47/d47Rco all(M1) /d47/d47Rco all(M2) /d47/d47Rco all(M3) inC-mod, where both rows are exact as Rco allis an exact functor, and the second and third columns are isomorphisms. Hence the first column is also an is omor- phism, and M1is complete, so C-modcois closed under kernels in C-mod. It is closed under cokernels and extensions by very similar arguments. 57Example 5.28. LetCbe aC∞-ring with ( X,OX) = Spec C. Then: (a)Considering Cas aC-module, we have Γ ◦MSpecC= Γ◦SpecC=OX(X), and Ψ C:C→OX(X) in Definitions 4.19 and 5.17 coincide. Hence Cis complete as a C-module if and only if it is complete as a C∞-ring, in the sense of§4.6. So, if Cis a finitely generated but not fair C∞-ring, as in Examples 2.19 and 2.21, then Cis a non-complete C-module. (b)Suppose Cis complete and Xis Lindel¨ of. Let Mbe a finitely presented C-module, so we have an exact sequence C⊗Rm→C⊗Rn→M→0 inC-mod. Here C⊗Rm,C⊗Rnare complete as Cis by(a), soMis complete by Proposition 5.27 as C-mod is closed under cokernels. (c)Suppose Cis complete, Xis Lindel¨ of, and I⊆Cis a finitely generated ideal. Choose generators i1,...,inforI. Then we have an exact sequence C⊗Rn→C→C/I→0 inC-mod with C⊗Rn,Ccomplete, so C/Iis a complete C-module by Proposition 5.27. Also we have an exact sequence 0→I→C→C/IwithC,C/Icomplete, so Iis a complete C-module. (d)LetCbe complete and Vbe an infinite-dimensional R-vector space. One can show that C⊗RVis a complete C-module if and only if Xis compact. 5.6 Cotangent sheaves of C∞-schemes We nowdefine cotangent sheaves , the sheafversionofcotangentmodules in §5.2. Definition 5.29. LetX= (X,OX) be aC∞-ringed space. Define PT∗Xto associate to each open U⊆Xthe cotangent module Ω OX(U)of Definition 5.3, regarded as a module over the C∞-ringOX(U), and to each inclusion of open setsV⊆U⊆Xthe morphism of OX(U)-modules Ω ρUV: ΩOX(U)→ΩOX(V) associated to the morphism of C∞-ringsρUV:OX(U)→OX(V). Then as we want for (5.13) the following commutes: OX(U)×ΩOX(U) ρUV×ΩρUV/d15/d15µOX(U)/d47/d47ΩOX(U) ΩρUV/d15/d15 OX(V)×ΩOX(V)µOX(V)/d47/d47ΩOX(V). Using this and functoriality of cotangent modules Ω ψ◦φ= Ωψ◦Ωφin Definition 5.3, we see thatPT∗Xis a presheaf ofOX-modules on X. Define the cotangent sheafT∗XofXto be the sheaf of OX-modules associated to PT∗X. IfU⊆Xis open then we have an equality of sheaves of OX|U-modules T∗(U,OX|U) =T∗X|U. As in Example 5.4, if f:X→Yis a smooth map of manifolds we have a morphism d f:f∗(T∗Y)→T∗Xof vector bundles over X. Here is an analogue forC∞-ringed spaces. Let f:X→Ybe a morphism of C∞-ringed spaces. Then by Definition 5.13, f∗(T∗Y) =f−1(T∗Y)⊗f−1(OY)OX,whereT∗Yis the 58sheafification of the presheaf V/ma√sto→ΩOY(V), andf−1(T∗Y) the sheafification of the presheaf U/ma√sto→limV⊇f(U)(T∗Y)(V), andf−1(OY) the sheafification of the presheafU/ma√sto→limV⊇f(U)OY(V). These three sheafifications combine into one, so thatf∗(T∗Y) is the sheafification of the presheaf P(f∗(T∗Y)) acting by U/ma√sto−→P(f∗(T∗Y))(U) = limV⊇f(U)ΩOY(V)⊗OY(V)OX(U). Define a morphism of presheaves PΩf:P(f∗(T∗Y))→PT∗XonXby (PΩf)(U) = limV⊇f(U)(Ωρf−1(V)U◦f♯(V))∗, where (Ω ρf−1(V)U◦f♯(V))∗: ΩOY(V)⊗OY(V)OX(U)→ΩOX(U)= (PT∗X)(U) is constructed as in Definition 5.3 from the C∞-ring morphisms f♯(V) :OY(V)→ OX(f−1(V)) fromf♯:OY→f∗(OX) corresponding to f♯infas in (4.3), and ρf−1(V)U:OX(f−1(V))→OX(U) inOX. Define Ω f:f∗(T∗Y)→T∗Xto be the induced morphism of the associated sheaves. Remark 5.30. There is an alternative definition of the cotangent sheaf T∗X following Hartshorne [31, p. 175]. We can form the product X×XinC∞RS, and there is a natural diagonal morphism ∆ X:X→X×X. WriteIXfor the sheaf of ideals in OX×Xvanishing on the closed C∞-ringed subspace ∆ X. ThenT∗X∼=∆∗ X(IX/I2 X). This can be proved using the equivalence of two definitions of cotangent module in [31, Prop. II.8.1A]. An affine version of this also appears in Dubuc and Kock [25]. Proposition 5.31. LetCbe aC∞-ring andX= SpecC. Then there is a canonical isomorphism T∗X∼=MSpecΩ C. Proof.By Definitions 5.16 and 5.29, MSpecΩ CandT∗Xare sheafifications of presheavesPMSpecΩ C,PT∗X, where for open U⊆Xwe have PMSpecΩ C(U) = ΩC⊗COX(U) andPT∗X(U) = ΩOX(U). We haveC∞-ringmorphismsΨ C:C→OX(X) fromDefinition 4.19andrestric- tionρXU:OX(X)→OX(U) fromOX, and so as in Definition 5.3 a morphism ofOX(U)-modulesPρ(U) := (ρXU◦ΨC)∗: ΩC⊗COX(U)→ΩOX(U). This de- fines a morphism of presheaves Pρ:PMSpecΩ C→PT∗X, and so sheafifying induces a morphism ρ: MSpecΩ C→T∗X. The induced morphism on stalks at x∈Xisρx= (πx)∗: ΩC⊗CCx→ΩCx, whereπx:C→Cxisprojectiontothelocal C∞-ringCx, notingthatOX,x∼=Cx. ButCxis the localization C[c−1:c∈C,c(x)/\e}atio\slash= 0], so Proposition 5.7 implies that (πx)∗: ΩC⊗CCx→ΩCxis an isomorphism. Hence ρ: MSpecΩ C→T∗X is a sheaf morphism which induces isomorphisms on stalks at all x∈X, soρis an isomorphism. Here are some properties of the morphisms Ω fin Definition 5.29. Equation (5.20) is an analogue of (5.6) and (5.12). 59Theorem 5.32. (a) Letf:X→Yandg:Y→Zbe morphisms of C∞- schemes. Then Ωg◦f= Ωf◦f∗(Ωg)◦If,g(T∗Z) (5.18) as morphisms (g◦f)∗(T∗Z)→T∗Xinqcoh(X). HereΩg:g∗(T∗Z)→T∗Yis a morphism in qcoh(Y),so applying f∗givesf∗(Ωg) :f∗(g∗(T∗Z))→f∗(T∗Y)in qcoh(X),andIf,g(T∗Z) : (g◦f)∗(T∗Z)→f∗(g∗(T∗Z))is as in Remark 5.14. (b)Suppose we are given a Cartesian square in C∞Sch Wf/d47/d47 e/d15/d15Y h/d15/d15 Xg/d47/d47Z,(5.19) so thatW=X×ZY. Then the following is exact in qcoh(W): (g◦e)∗(T∗Z)e∗(Ωg)◦Ie,g(T∗Z)⊕ −f∗(Ωh)◦If,h(T∗Z)/d47/d47e∗(T∗X) ⊕f∗(T∗Y)Ωe⊕Ωf/d47/d47T∗W /d47/d470.(5.20) Proof.Combining several sheafifications into one as in the proof of Propos ition 5.24, we see that the sheaves T∗X,f∗(T∗Y),f∗(g∗(T∗Z)) and (g◦f)∗(T∗Z) on Xare isomorphic to the sheafifications of the following presheaves: T∗X/squigglerightU/ma√sto−→ΩOX(U), (5.21) f∗(T∗Y)/squigglerightU/ma√sto−→lim V⊇f(U)ΩOY(V)⊗OY(V)OX(U), (5.22) f∗(g∗(T∗Z))/squigglerightU/ma√sto−→lim V⊇f(U)lim W⊇g(V)/parenleftbig ΩOZ(W)⊗OZ(W)OY(V)/parenrightbig ⊗OY(V)OX(U),(5.23) (g◦f)∗(T∗Z)/squigglerightU/ma√sto−→lim W⊇g◦f(U)ΩOZ(W)⊗OZ(W)OX(U). (5.24) Then Ωf,Ωg◦f,f∗(Ωg),If,g(T∗Z) are the morphisms of sheaves associated to the following morphisms of the presheaves in (5.21)–(5.24): Ωf/squigglerightU/ma√sto−→lim V⊇f(U)(Ωρf−1(V)U◦f♯(V))∗, (5.25) Ωg◦f/squigglerightU/ma√sto−→lim W⊇g◦f(U)(Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗,(5.26) f∗(Ωg)/squigglerightU/ma√sto−→lim V⊇f(U)lim W⊇g(V)(Ωρg−1(W)V◦g♯(W))∗,(5.27) If,g(T∗Z)/squigglerightU/ma√sto−→lim V⊇f(U)lim W⊇g(V)IUVW, (5.28) by Definition 5.29, where IUVW: ΩOZ(W)⊗OZ(W)OX(U)→/parenleftbig ΩOZ(W)⊗OZ(W) OY(V)/parenrightbig ⊗OY(V)OX(U) is the natural isomorphism. Now ifU⊆X,V⊆Y,W⊆Zare open with V⊇f(U),W⊇g(V) then ρ(g◦f)−1(W)U◦(g◦f)♯(W) =/bracketleftbig ρf−1(V)U◦f♯(V)/bracketrightbig ◦/bracketleftbig ρg−1(W)V◦g♯(W)/bracketrightbig 60as morphismsOZ(W)→OX(U), so Ωφ◦ψ= Ωφ◦Ωψin Definition 5.3 implies (Ωρ(g◦f)−1(W)U◦(g◦f)♯(W))∗= (Ωρf−1(V)U◦f♯(V))∗◦(Ωρg−1(W)V◦g♯(W))∗◦IUVW. Taking limits lim V⊇f(U)limW⊇g(V)implies that the morphisms of presheaves in (5.25)–(5.28) satisfy the analogue of (5.18), so passing to sheave s proves (a). For (b), first observe that as (5.19) is commutative, by (a) we hav e Ωe◦e∗(Ωg)◦Ie,g(T∗Z) = Ωg◦e= Ωh◦f= Ωf◦f∗(Ωh)◦If,h(T∗Z), so Ωe◦/parenleftbig e∗(Ωg)◦Ie,g(T∗Z)/parenrightbig −Ωf◦/parenleftbig f∗(Ωh)◦If,h(T∗Z)/parenrightbig = 0, and (5.20) is a complex. To show it is exact, note that as in the first pa rt of the proof, (5.20) is the sheafification of a complex of presheave s, and the presheaves are defined as direct limits. Let S⊆Wbe open. Then the complex ofpresheavescorrespondingto (5.20) evaluatedat S⊆Wis the directlimit over all openT⊆X,U⊆Y,V⊆Zwithe(S)⊆T,f(S)⊆U,g(T)⊆V,h(U)⊆V of equation (5.6) with OZ(V),OX(T),OY(U),OW(S) in place of C,D,E,F. Since (5.6) is exact by Theorem 5.8 and direct limits are exact, the com plex ofpresheaveswhose sheafificationis (5.20) is exact when evaluate d on each open S⊆W, so it is exact. As sheafification is an exact functor, this implies that equation (5.20) is exact. This completes the proof. 6C∞-stacks We now discuss C∞-stacks, that is, geometric stacks over the site ( C∞Sch,J) ofC∞-schemes with the open cover topology. The author knows of no pr evious work on these. For the rest of the book, we will assume the reader has some familiarity with stacks in algebraicgeometry. Appendix A summarizes t he main definitions and results on stacks that we will use, but it is too brief to help someone learn about stacks for the first time. Readers with little ex perience of stacks are advised to first consult an introductory text such a s Vistoli [68], Gomez [29], Laumon and Moret-Bailly [46], or the online ‘Stacks Project ’ [34]. The author found Metzler [49] and Noohi [55] useful in writing this section. 6.1C∞-stacks We use the material of §A.2–§A.5. Definition 6.1. Define a Grothendieck pretopology PJon the category of C∞-schemes C∞Schto have coverings {ia:Ua→U}a∈AwhereVa=ia(Ua) is open inUwithia:Ua→(Va,OU|Va) and isomorphism for all a∈A, and U=/uniontext a∈AVa. Using Corollary 4.29 we see that up to isomorphisms of the Ua, the coverings{ia:Ua→U}a∈AofUcorrespond exactly to open covers {Va:a∈A}ofU. WriteJfor the associated Grothendieck topology. It is a straightforward exercise in sheaf theory to prove: 61Proposition6.2. The site(C∞Sch,J)has descent for objects and morphisms, in the sense of§A.3. Thus it is subcanonical. The point here is that since coverings of UinJare just open covers of the underlying topological space U, rather than something more complicated like ´ etale covers in algebraic geometry, proving descent is easy: for o bjects, we glue the topological spaces XaofXatogether in the usual way to get a topological spaceX, then we glue the OXatogether to get a presheaf of C∞-rings˜OXon Xisomorphic toOXaonXa⊆Xfor alla∈A, and finally we sheafify ˜OXto a sheaf ofC∞-ringsOXonX, which is still isomorphic to OXaonXa⊆X. Definition 6.3. AC∞-stackXis a geometric stack on the site ( C∞Sch,J). WriteC∞Stafor the 2-category of C∞-stacks,C∞Sta=GSta(C∞Sch,J). As in Definition A.13, we will very often use the notation that if Xis a C∞-scheme then ¯Xis the associated C∞-stack, and if f:X→Yis a mor- phism ofC∞-schemes then ¯f:¯X→¯Yis the associated 1-morphism of C∞- stacks. Write ¯C∞Schlfp,¯C∞Schlf,¯C∞Schfor the full 2-subcategories of C∞- stacksXinC∞Stawhich are equivalent to ¯XforXinC∞Schlfp,C∞Schlf orC∞Sch, respectively. When we say that a C∞-stackXis aC∞-scheme, we mean thatX∈¯C∞Sch. Since(C∞Sch,J)isasubcanonicalsite, theembedding C∞Sch→C∞Sta takingX/ma√sto→¯X,f/ma√sto→¯fis fully faithful. We write this as a full and faithful functorFC∞Sta C∞Sch:C∞Sch→C∞StamappingFC∞Sta C∞Sch:X/ma√sto→¯Xon objects andFC∞Sta C∞Sch:f/ma√sto→¯fon (1-)morphisms. Hence ¯C∞Schlfp,¯C∞Schlf,¯C∞Sch areequivalentto C∞Schlfp,C∞Schlf,C∞Sch, consideredas 2-categorieswith only identity 2-morphisms. In practice one often does not distinguis h between schemes and stacks which are equivalent to schemes, that is, one id entifies C∞Schlfp,...,C∞Schand¯C∞Schlfp,...,¯C∞Sch. Remark 6.4. Behrend and Xu [5, Def. 2.15] use ‘ C∞-stack’to mean something different, a stack Xover the site ( Man,JMan) of manifolds with Grothendieck topologyJManassociated to the Grothendieck pretopology PJMangiven by opencovers,suchthatthereexistsasurjectiverepresentable submersion π:¯U→ Xfrom some manifold U. These arealsocalled ‘smooth stacks’or‘differentiable stacks’in[5,32,49,55]. Thequotient[ V/G]ofamanifold VbyaLiegroup Gisan example of a differentiable stack. By Zung’s linearization theorem [71 , Th. 2.3], a differentiable stack Xwith proper diagonal is Zariski locally equivalent to such a quotient [ V/G] withGcompact. Our C∞-stacks are a far larger class of more singular objects than the differentiable stacks of [5,32,49, 55]. Theorems 4.25(b) and A.23, Corollary A.26 and Proposition 6.2 imply: Theorem 6.5. LetXbe aC∞-stack. ThenXis equivalent to the stack [V⇒U]associated to a groupoid (U,V,s,t,u,i,m)inC∞Sch. Conversely, any groupoid in C∞Schdefines aC∞-stack[V⇒U]. All fibre products exist in the2-category C∞Sta. QuotientC∞-stacks[X/G] are a special class of C∞-stacks. 62Definition 6.6. AC∞-groupGis a group object in C∞Sch, that is, a C∞- schemeG= (G,OG) equipped with an identity element 1 ∈Gand multiplica- tion and inverse morphisms m:G×G→G,i:G→GinC∞Schsuch that (∗,G,π,π,1,i,m) is a groupoid in C∞Sch. Here∗= SpecRis a point, and π:G→∗is the projection, and we regard 1 ∈Gas a morphism 1 : ∗→G. LetGbe aC∞-group, and XaC∞-scheme. A ( left)actionofGonXis a morphismµ:G×X→Xsuch that /parenleftbig X,G×X,πX,µ,1×idX,(i◦πG)×µ,(m◦((πG◦π1)×(πG◦π2)))×(πX◦π2)/parenrightbig (6.1) is a groupoid object in C∞Sch, where in the final morphism π1,π2are the projections from ( G×X)×πX,X,µ(G×X) to the first and second factors G×X. Then define the quotientC∞-stack[X/G] to be the stack [ G×X⇒X] associated to the groupoid (6.1). It is a C∞-stack. IfG= (G,OG) is aC∞-group then the underlying space Gis a topological group, and is in particular a group, and if G= (G,OG) acts onX= (X,OX) thenGacts continuously on X. IfGis a Lie group then G=FC∞Sch Man(G) is aC∞-group in a natural way, by applyingFC∞Sch Mantothesmoothmultiplicationandinversemaps m:G×G→G andi:G→G. If a Lie group Gacts smoothly on a manifold Xwith action µ:G×X→Xthen theC∞-groupG=FC∞Sch Man(G) acts on the C∞-scheme X=FC∞Sch Man(X) with action µ=FC∞Sch Man(µ) :G×X→X, so we can form the quotient C∞-stack [X/G]. Example 6.7. LetGbe aC∞-group, and X=∗be the point in C∞Sch, with trivialG-action. The quotient C∞-stack [∗/G] is known as BG, the classifying stack for principal G-bundles on C∞-schemes. IfSis aC∞-scheme, a principalG-bundle(P,π,µ) overSis aC∞-scheme P, a morphism π:P→S, and aG-actionµ:G×P→PofGonP, such thatπisG-invariant, and Smay be covered by open C∞-subschemes U⊆S such that there exists an isomorphism π−1(U)∼=G×Uwhich identifies the G-action onπ−1(U)⊆Pwith the product of the left G-action onGand the trivialG-action onU, and identifies π|···:π−1(U)→UwithπU:G×U→U. Often we write Pas the principal bundle, leaving π,µimplicit. One well known way to write BGexplicitly as a category fibred in groupoids pX:X→C∞Sch, as in§A.2, is to defineXto be the category with objects pairs (S,P) of aC∞-schemeSandPa principal G-bundle over S, and mor- phisms (f,u) : (S,P)→(T,Q) consisting of C∞-scheme morphisms f:S→T andu:P→Q, such that uisG-equivariant and P π/d15/d15u/d47/d47Q π/d15/d15 Sf/d47/d47T(6.2) is a Cartesian square in C∞Sch, which implies that Pis canonicallyisomorphic to the pullback principal G-bundlef∗(Q). Composition of morphisms is ( g,v)◦ 63(f,u) = (g◦f,v◦u), and identity morphisms are id (S,P)= (idS,idP). The functorpX:X→C∞SchmapspX: (S,P)/ma√sto→Sonobjectsand pX: (f,u)/ma√sto→f on morphisms. In§7.1 we will give a more detailed treatment of quotient C∞-stacks [X/G] of aC∞-schemeXby a finite group G. 6.2 Properties of 1-morphisms of C∞-stacks We use the material of §A.4. We define some classes of C∞-scheme morphisms. Definition 6.8. Letf= (f,f♯) :X= (X,OX)→Y= (Y,OY) be a morphism inC∞Sch. Then: •We callfanopen embedding ifV=f(X) is an open subset in Yand (f,f♯) : (X,OX)→(V,OY|V) is an isomorphism. •We callfaclosed embedding iff:X→Yis a homeomorphism with a closed subset of Y, andf♯:f−1(OY)→OXis a surjective morphism of sheaves of C∞-rings. Equivalently, fis an isomorphism with a closed C∞-subscheme of Y. Over affine open subsets U∼=SpecCinY,fis modelled on the natural morphism Spec( C/I)֒→SpecCfor some ideal I inC. •We callfanembedding if we may write f=g◦hwherehis an open embedding and gis a closed embedding. •We callf´ etaleif eachx∈Xhas an open neighbourhood UinXsuch thatV=f(U) is open in Yand (f|U,f♯|U) : (U,OX|U)→(V,OY|V) is an isomorphism. That is, fis a local isomorphism. •We callfproperiff:X→Yis a proper map of topological spaces, that is, ifS⊆Yis compact then f−1(S)⊆Xis compact. •We say that fhas finite fibres iff:X→Yis a finite map, that is, f−1(y) is a finite subset of Xfor ally∈Y. •We callfseparated iff:X→Yis a separated map of topological spaces, that is, ∆ X=/braceleftbig (x,x) :x∈X/bracerightbig is a closed subset of the topological fibre productX×f,Y,fX=/braceleftbig (x,x′)∈X×X:f(x) =f(x′)/bracerightbig . •We callfclosediff:X→Yis a closed map of topological spaces, that is,S⊆Xclosed implies f(S)⊆Yclosed. •We callfuniversally closed if whenever g:W→Yis a morphism then πW:X×f,Y,gW→Wis closed. •We callfasubmersion if for allx∈Xwithf(x) =y, there exists an open neighbourhood UofyinYand a morphism g= (g,g♯) : (U,OY|U)→ (X,OX) withg(y) =xandf◦g= id(U,OY|U). 64•We callflocally fair , orlocally finitely presented , if whenever Uis a locally fair, or locally finitely presented C∞-scheme, respectively, and g:U→Y is a morphism then X×f,Y,gUis locally fair, or locally finitely presented, respectively. Remark 6.9. These are mostly analogues of standard concepts in algebraic geometry, as in Hartshorne [31] for instance. But because the to pology onC∞- schemes is finer than the Zariski topology in algebraic geometry — fo r example, affineC∞-schemes are Hausdorff — our definitions of ´ etale and proper are s im- pler than in algebraic geometry. (Open or closed) embeddings corre spond to (open or closed) immersions in algebraic geometry, but we prefer th e word ‘em- bedding’, as immersion has a different meaning in differential geometry . Closed morphisms are not invariant under base change, which is why we defin e univer- sally closed. If X,Yare manifolds and X,Y=FC∞Sch Man(X,Y), thenf:X→Y is a submersion of C∞-schemes if and only if f=FC∞Sch Man(f) forf:X→Ya submersion of manifolds. Definition 6.10. LetPbe a property of morphisms in C∞Sch. We say that Pis stable under open embedding if whenever f:U→VisPandi:V→W is an open embedding, then i◦f:U→WisP. The next proposition is elementary. See Laumon and Bailly [46, §3.10] and Noohi [55, Ex. 4.6] for similar lists for the ´ etale and topological sites . Proposition 6.11. The following properties of morphisms in C∞Schare in- variant under base change and local in the target in the site (C∞Sch,J),in the sense of§A.4:open embedding, closed embedding, embedding, ´ etale, prop er, has finite fibres, separated, universally closed, submersio n, locally fair, locally finitely presented. The following properties are also stabl e under open embed- ding, in the sense of Definition 6.10:open embedding, embedding, ´ etale, has finite fibres, separated, submersion, locally fair, locally finitely presented. As in§A.4, this implies that these properties are also defined for repre- sentable 1-morphisms in C∞Sta. In particular, if Xis aC∞-stack then ∆ X: X →X×X is representable, and if Π : ¯U→Xis an atlas then Π is repre- sentable, so we can require that ∆ Xor Π has some of these properties. Definition 6.12. LetXbe aC∞-stack. Following [46, Def. 7.6], we say that X isseparated if the diagonal 1-morphism ∆ X:X→X×X is universally closed. IfX=¯Xfor someC∞-schemeX= (X,OX) thenXis separated if and only if ∆X:X→X×Xis closed, that is, if and only if Xis Hausdorff. Proposition 6.13. LetW=X×f,Z,gYbe a fibre product of C∞-stacks with X,Yseparated. ThenWis separated. 65Proof.We have a 2-commutative diagram with both squares 2-Cartesian: W∆W/d47/d47 /d117/d117❦❦❦❦❦❦❦❦❦❦❦π1/d41/d41❙❙❙❙❙❙❙❙❙ W×W /d41/d41❙❙❙❙❙❙❙ Z Z /d41/d41❙❙❙❙❙❙❙❙❙❙❙ X×f◦∆Z,Z×Z,g◦ZY /d117/d117❦❦❦❦❦❦❦❦ /d41/d41❙❙❙❙❙❙π2/d53/d53❦❦❦❦❦❦❦ X×X×Y×Y . IZ X×Y∆X×∆Y/d53/d53❦❦❦❦❦❦❦(6.3) Let [V⇒U] be a groupoid presentation of Z, and consider the fourth 2- Cartesian diagram of (A.12), with surjective rows. The left hand mo rphism ¯uׯidUhas a left inverse πU, and so is automatically universally closed. Hence Zis universally closed by Propositions A.18(c) and 6.11, so π1in (6.3) is uni- versally closed by Propositions A.18(a) and 6.11. Also ∆ X,∆Yare universally closed asX,Yare separated, so ∆ X×∆Yin (6.3) is universally closed, and π2is universally closed. Thus ∆ W∼=π2◦π1is universally closed, and Wis separated. 6.3 Open C∞-substacks and open covers Definition 6.14. LetXbe aC∞-stack. AC∞-substackYinXis a substack ofX, in the sense of Definition A.7, which is also a C∞-stack. It has a natural inclusion 1-morphism iY:Y֒→X. We callYanopenC∞-substack ofXif iYis a representable open embedding, a closedC∞-substack ofXifiYis a representable closed embedding, and a locally closed C∞-substack ofXifiYis a representable embedding. Anopen cover{Ua:a∈A}ofXis a family of open C∞-substacksUainX with/coproducttext a∈AiUa:/coproducttext a∈AUa→Xsurjective. We write U⊆XwhenUis an open C∞-substack ofX, and/uniontext a∈AU=Xto mean that/coproducttext a∈AiUais surjective. Somepropertiesof∆ X,ιX,XandatlasesforXcanbetestedontheelements of an open cover. The proof is elementary. Proposition 6.15. LetXbe aC∞-stack, and{Ua:a∈A}an open cover ofX. Suppose PandQare properties of morphisms in C∞Schwhich are invariant under base change and local in the target in (C∞Sch,J),and that P is stable under open embedding. Then: (a)LetΠa:¯Ua→Uabe an atlas forUafora∈A. SetU=/coproducttext a∈AUaand Π =/coproducttext a∈AiUa◦Πa:¯U→X. ThenΠis an atlas forX,andΠisPif and only if ΠaisPfor alla∈A. (b)∆X:X→X×X isPif and only if ∆Ua:Ua→Ua×UaisPfor alla∈A. (c)ιX:IX→XisQif and only if ιUa:IUa→UaisQfor alla∈A. (d)X:X→IXisQif and only if Ua:Ua→IUaisQfor alla∈A. IfX=¯Ufor someC∞-schemeU= (U,OU), then the open C∞-substacks ofXare precisely those subsheaves of the form (V,OU|V) for all open V⊆U, that is, they are the images in C∞Staof the open C∞-subschemes of U. We can also describe the open substacks of stacks [ V⇒U] associated to groupoids: 66Proposition 6.16. Let(U,V,s,t,u,i,m)be a groupoid in C∞SchandX= [V⇒U]the associated C∞-stack, and write U= (U,OU),and so on. Then open C∞-substacksX′ofXare naturally in 1-1correspondence with open subsets U′⊆Uwiths−1(U′) =t−1(U′),whereX′= [V′⇒U′]forU′= (U′,OU|U′) andV′= (s−1(U′),OV|s−1(U′)). If(U,V,s,t,u,i,m)is as in(6.1),so thatX is a quotient C∞-stack[U/G],then openC∞-substacksX′ofXcorrespond to G-invariant open subsets U′⊆U. Proof.From Theorem A.23, as X= [V⇒U] we have a natural surjective, representable 1-morphism Π : ¯U→X. IfX′is an openC∞-substack ofXthen ¯U×Π,X,iX′X′is an open C∞-substack of ¯U, and so is of the form (U′,OU|U′) for some open U′⊆U. We have natural equivalences (s−1(U′),OV|s−1(U′))≃¯U′×i¯U′,¯U,¯s¯V≃X′×X(¯U×id¯U,¯U,¯s¯V)≃X′×i′ X,X,πX¯V ≃X′×X(¯U×id¯U,¯U,¯t¯V)≃¯U′×i¯U′,¯U,¯t¯V≃(t−1(U′),OV|t−1(U′)), by associativity properties of fibre products in 2-categories, whic h implies that s−1(U′) =t−1(U′). Conversely, if s−1(U′) =t−1(U′) then defining U′,V′as in the proposition, we get a C∞-stackX′= [V′⇒U′] which is naturally an open C∞-substack ofX. WhenX= [U/G], we see that s−1(U′) =t−1(U′) if and only ifU′isG-invariant. 6.4 The underlying topological space of a C∞-stack Following Noohi [55, §4.3,§11] in the case of topological stacks, we associate a topological space Xtopto aC∞-stackX. In§7.4, ifXis a Deligne–Mumford C∞-stack, we will also give Xtopthe structure of a C∞-scheme. Definition 6.17. LetXbe aC∞-stack. Write∗for the point Spec Rin C∞Sch, and¯∗for the associated point in C∞Sta. DefineXtopto be the set of 2-isomorphism classes [ x] of 1-morphisms x:¯∗→X. SupposeU⊆Xis an openC∞-substack. SinceUis a subcategory of X, any 1-morphism u:¯∗→U, regardedasafunctorfromthecategory ¯∗tothe category U, is also a 1-morphism u:¯∗→X. Also, asUis a strictly full subcategory of X, ifx:¯∗→Xis a 1-morphism and η:u⇒xa 2-morphism of 1-morphisms ¯∗→X, thenxis also a 1-morphism u:¯∗→U, andηis also a 2-morphism of 1-morphisms ¯∗→U. This implies that Utopis a subset ofXtop. DefineTXtop=/braceleftbig Utop:U⊆XisanopenC∞-substackinX/bracerightbig , asetofsubsets ofXtop. We claimthatTXtopisatopologyonXtop. Toseethis, notethattaking Uto beXor the empty C∞-substack givesXtop,∅∈TXtop. IfU,V⊆Xare openC∞-substacks ofXthen the intersection of subcategories W=U∩Vis an openC∞-substack ofXequivalent to the fibre product U×iU,X,iVV, with Wtop=Utop∩Vtop, soTXtopis closed under finite intersections. If{Ua:a∈A}is a family of open C∞-substacks inX, defineVto be the unique smallest strictly full subcategory of Xwhich containsUafor eacha∈A and is closed under the stack axiom (A.9) in Definition A.6. Then Vis an open 67C∞-substack ofX, which we write as V=/uniontext a∈AUa, andVtop=/uniontext a∈AUatop. SoTXtopis closed under arbitrary unions. Thus (Xtop,TXtop) is a topological space, which we call the underlying topo- logical space ofX, and usually write as Xtop. It has the following properties. Iff:X →Y is a 1-morphism of C∞-stacks then there is a natural continu- ous mapftop:Xtop→Ytopdefined by ftop([x]) = [f◦x]. Iff,g:X →Y are 1-morphisms and η:f⇒gis a 2-isomorphism then ftop=gtop. Map- pingX /ma√sto→X top,f/ma√sto→ftopand 2-morphisms to identities defines a 2-functor FTop C∞Sta:C∞Sta→Top, where the category of topological spaces Topis regarded as a 2-category with only identity 2-morphisms. IfX= (X,OX) is aC∞-scheme, so that ¯Xis aC∞-stack, then ¯Xtopis naturallyhomeomorphicto X, and we will identify ¯XtopwithX. Iff= (f,f♯) : X= (X,OX)→Y= (Y,OY) is a morphism of C∞-schemes, so that ¯f:¯X→¯Y is a 1-morphism of C∞-stacks, then ¯ftop:¯Xtop→¯Ytopisf:X→Y. For aC∞-stackX, we can characterize Xtopby the following universal property. We are given a topological space Xtopand for every 1-morphism f:¯U→Xfor aC∞-schemeU= (U,OU) we are given a continuous map ftop:U→Xtop, such that if fis 2-isomorphic to h◦¯gfor some morphism g= (g,g♯) :U→Vand 1-morphism h:V→Xthenftop=htop◦g. IfX′ top, f′ topare alternative choices of data with these properties then there is a unique continuous map j:Xtop→X′ topwithf′ top=j◦ftopfor allf. We can also make Xtopinto aC∞-ringed spaceXtop: Definition 6.18. LetXbe aC∞-stack. Define a sheaf of C∞-ringsOXtop onXtopas follows: each open set in XtopisUtopfor some unique open C∞- substackU ⊆X. DefineOXtop(Utop) to be the set of 2-isomorphism classes [c] of 1-morphisms c:U →¯R. Iff:Rn→Ris smooth and [ c1],...,[cn]∈ OXtop(Utop), define Φ f/parenleftbig [c1],...,[cn]/parenrightbig =/bracketleftbig¯f◦(c1×···×cn)/bracketrightbig , using the compo- sitionUc1×···×cn−→¯R×···×¯R¯f−→¯R. ThenOXtop(Utop) is aC∞-ring. IfVtop⊆Utop⊆Xtopare open, so that V ⊆U ⊆X , define aC∞-ring morphismρUV:OXtop(Utop)→OXtop(Vtop) byρUV: [c]/ma√sto→[c|V]. It is now easy to check that OXtopis a presheaf of C∞-rings onXtop, but it is less obvious that it is a sheaf. To see this, note that by general proper ties of stacks, U /ma√sto→Hom(U,¯R) is a 2-sheaf (stack) of groupoids on the topological space Xtop, whereHom(U,¯R) is the groupoid of 1- and 2-morphisms U →¯R, and OXtop(Utop) is its set of isomorphism classes. Starting with a 2-sheaf and taking sets of isomorphism classes gene rally yields only a presheaf of sets, not a sheaf. But as ¯Ris aC∞-scheme the groupoids Hom(U,¯R) are discrete (have no nontrivial automorphisms), so tak- ing isomorphism classes loses no information, and the 2-sheaf prope rty implies thatOXtopis a sheaf of sets, and so of C∞-rings. ThusXtop= (Xtop,OXtop) is aC∞-ringed space, the underlying C∞-ringed space ofX. For generalXthisXtopneed not be a C∞-scheme. If it is, we call Xtopthe coarse moduli C∞-scheme ofX. Coarse moduli C∞-schemes have the following universal property: there is a 1-morphism π:X →¯Xtopcalled the structural 68morphism , such that if f:X→¯Yis a 1-morphism for any C∞-schemeYthen fis 2-isomorphic to ¯ g◦πfor some unique C∞-scheme morphism g:Xtop→Y. We can think of a C∞-stackXas being a topological space Xtopequipped with some complicated extra geometrical structure, just as manif olds and orb- ifolds are usually thought of as topological spaces equipped with ext ra structure coming from an atlas of charts. As in Noohi [55, Ex. 4.13], it is easy to d escribe Xtopusing a groupoid presentation [ V⇒U] ofX: Proposition 6.19. LetXbe equivalent to the C∞-stack[V⇒U]associated to a groupoid (U,V,s,t,u,i,m)inC∞Sch,whereU= (U,OU),s= (s,s♯),and so on. Define∼onUbyp∼p′if there exists q∈Vwiths(q) =pandt(q) =p′. Then∼is an equivalence relation on U,so we can form the quotient U/∼,with the quotient topology. There is a natural homeomorphism Xtop∼=U/∼. For a quotient C∞-stackX≃[U/G]we haveXtop∼=U/G. Using this we can deduce properties of Xtopfrom properties of Xexpressed interms ofV⇒U. Forinstance, ifXis separatedthen s×t:V→U×Uis (uni- versally) closed, and we can take UHausdorff. But the quotient of a Hausdorff topological space by a closed equivalence relation is Hausdorff, yieldin g: Lemma 6.20. LetXbe a separated C∞-stack. Then the underlying topological spaceXtopis Hausdorff. Next we discuss isotropy groups ofC∞-stacks. Definition 6.21. LetXbe aC∞-stack, and [ x]∈Xtop. Pick a representative xfor [x], so thatx:¯∗→Xis a 1-morphism. Then there exists a C∞-scheme G= (G,OG), unique up to isomorphism, with ¯G=¯∗×x,X,x¯∗. Applying the construction of the groupoid in Definition A.21 with Π : U→Xreplaced by x:¯∗→X, we giveGthe structure of a C∞-group. The underlying group Gis canonically isomorphic to the group of 2-morphisms η:x⇒x. With [x]fixed, this C∞-groupGisindependent ofchoicesuptononcanonical isomorphism; roughly, Gis canonical up to conjugation in G. We define the isotropy group (ororbifold group , orstabilizer group )IsoX([x])orIso([x])of[x]to be thisC∞-groupG, regarded as a C∞-group up to noncanonical isomorphism. IfX= [V⇒U] is associatedto a groupoid( U,V,s,t,u,i,m) thenx:¯∗→X factors through ¯ w:¯∗→¯Uup to 2-isomorphism for some point w∈U, and then Gis isomorphic to the C∞-subscheme G′=s−1(w)∩t−1(w) inV, with identity u|w:∗→G′, inversei|G′:G′→G′, and multiplication m|G′×G′:G′×G′→G′. Iff:X→Yis a 1-morphism of C∞-stacks and [ x]∈Xtopwithftop([x]) = [y]∈Ytop, fory=f◦x, then at the level of sets we define f∗: IsoX([x])→ IsoY([y]) byf∗(η) = idf∗η. This is a group morphism, by compatibility of horizontal and vertical composition in 2-categories. We can exten df∗naturally to a morphism f∗: IsoX([x])→IsoY([y]) ofC∞-groups, such that ¯f∗:IsoX([x]) =¯∗×x,X,x¯∗−→¯∗×f◦x,Y,f◦x¯∗=IsoY([y]) is induced from f:X→Yby the universal property of fibre products. Then f∗,f∗are independent of the choice of x∈[x] up to conjugation in Iso Y([y]). 696.5 Gluing C∞-stacks by equivalences Here are two propositions on gluing C∞-stacks by equivalences. They are exer- cises in stack theory, with no special C∞issues, and also hold for other classes of stacks. See Rydh [61, Th. C] for stronger results for algebraic stacks. Proposition 6.22. SupposeX,YareC∞-stacks,U ⊆X,V ⊆Yare open C∞-substacks, and f:U→Vis an equivalence in C∞Sta. Then there exist aC∞-stackZ,openC∞-substacks ˆX,ˆYinZwithZ=ˆX∪ˆY,equivalences g:X →ˆXandh:Y→ˆYsuch thatg|Uandh|Vare both equivalences with ˆX∩ˆY,and a2-morphism η:g|U⇒h◦f:U→ˆX∩ˆYinC∞Sta. Furthermore, Zis independent of choices up to equivalence. Proposition 6.23. SupposeX,YareC∞-stacks,U,V ⊆ X are openC∞- substacks withX=U∪V, f:U → Y andg:V → Y are1-morphisms, andη:f|U∩V⇒g|U∩Vis a2-morphism in C∞Sta. Then there exists a 1- morphismh:X →Y and2-morphisms ζ:h|U⇒f, θ:h|V⇒gsuch that θ|U∩V=η⊙ζ|U∩V:h|U∩V⇒g|U∩V. Thishis unique up to 2-isomorphism. In general, hisnotindependent up to 2-isomorphism of the choice of η. Here is an example in which his not independent of ηup to 2-isomorphism in the last part of Proposition 6.23. Example 6.24. LetXbe theC∞-stack associated to the circle X=/braceleftbig (x,y)∈ R2:x2+y2= 1/bracerightbig , andU,V⊆Xthe substacks associated to the open sets U=/braceleftbig (x,y)∈X:x >−1 2/bracerightbig andV=/braceleftbig (x,y)∈X:x <1 2/bracerightbig . LetYbe the quotient C∞-stack [∗/Z2]. Then 1-morphisms f:X →Y correspond to principal Z2- bundlesPf→X, and for 1-morphisms f,g:X→Ywith principal Z2-bundles Pf,Pg→X,a2-morphism η:f⇒gcorrespondstoanisomorphismofprincipal Z2-bundlesPf∼=Pg. The same holds for 1-morphisms U,V,U∪V→Y and their 2-morphisms. Letf:U → Y andg:V → Y be the 1-morphisms corresponding to the trivial Z2-bundlesPf=Z2×U→U,Pg=Z2×V→V. Then 2- morphisms η:f|U∩V⇒g|U∩Vcorrespond to automorphisms of the trivial Z2-bundleZ2×(U∩V)→U∩V, that is, to continuous maps U∩V→Z2. Note thatU∩Vhas two connected components/braceleftbig (x,y)∈X:−1 2< x <1 2, y>0/bracerightbig and/braceleftbig (x,y)∈X:−1 2