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Battle For 2019 : NDA Finalises Seat Sharing Arrangement In Bihar, It is 17-17- 6
After days of intense deliberations and pressure from allies, the NDA has announced the seat sharing arrangement in Bihar. According to the media reports BJP – JD ( U) would contest 17 seats each while LJP will contest 6 . Bihar has 40 Lok Sabha seats.
The deal was announced by BJP president Amit Shah, in the presence of Bihar Chief Minister Nitish Kumar, BJP General Secretary Bhupender Yadav, Ram Vilas Paswan and his son Chirag Paswan
Besides the six Lok Sabha seats LJP leader Ram Vilas Paswan , who is the minister in the Modi government is likely to get a Rajya Sabha seat from either, Uttar Pradesh , Jharkhand .
BJP had contested 22 seats last time and winning them in 2014 .
The JD(U), which had fought the 2014 polls independently, and the LJP, which was a BJP ally, had won two and six seats, respectively. | {
"pile_set_name": "OpenWebText2"
} |
Check out our new site Makeup Addiction
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I found the perfect position in bed and immediately realized i forgot to brush my teeth | {
"pile_set_name": "OpenWebText2"
} |
/*
* Copyright 2011 X-Scale. All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY X-Scale ``AS IS'' AND ANY EXPRESS OR IMPLIED
* WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO
* EVENT SHALL X-Scale OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
* OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
* LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
* NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE,
* EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* The views and conclusions contained in the software and documentation are
* those of the authors and should not be interpreted as representing official
* policies, either expressed or implied, of X-Scale.
*
* This software provides an algorithm that emulates the protection scheme of
* N64 PIF/CIC-NUS-6105, by determining the proper response to each challenge.
* It was synthesized after a careful, exhaustive and detailed analysis of the
* challenge/response pairs stored in the 'pif2.dat' file from Project 64.
* These challenge/response pairs were the only resource used during this
* project. There was no kind of physical access to N64 hardware.
*
* This project would have never been possible without the contribuitions of
* the following individuals and organizations:
*
* - Oman: For being at the right place at the right time and being brave
* enough to pay a personal price so we could understand in a much deeper
* way how this magical console really works. We owe you so much.
*
* - Jovis: For all the positive energy and impressive hacking spirit that you
* shared with the N64 community. You were absolutely instrumental in
* several key events that shaped the N64 community in the last 14 years.
* Even if you're not physically with us anymore, your heritage, your
* knowledge and your attitude will never be forgotten.
*
* 'The candle that burns twice as bright burns half as long.'
*
* - LaC: For the endless contributions that you've given to the N64 community
* since the early days, when N64 was the next big thing. I've always
* admired the deep knowledge that you've gathered about the most little
* hardware details. Recently, you challanged us to find a small and
* concise algorithm that would emulate the behaviour of CIC-NUS-6105
* challenge/response protection scheme and here is the final result.
* LaC, Oman and Jovis were definitly the dream team of N64 reversing in
* the late 90's. Without your contributions, we would be much poorer.
*
* - marshall: For keeping the N64 scene alive during the last decade, when
* most people lost interest and moved along to different projects. You
* are the force that has been keeping us all together in the later
* years. When almost nobody cared about N64 anymore, you were always
* there, spreading the word, developing in the console, and lately,
* making impressive advances on the hardware side. I wish the best
* success to your new 64drive project.
*
* - hcs: For your contributions to the better understanding of the inner
* workings of the Reality Co-Processor (RCP). Your skills have impressed
* me for a long time now. And without your precious help by sharing your
* kownledge, I would have never understood the immense importance of
* Oman, Jovis and LaC achievements. Thank you !
*
* - Azimer & Tooie: For sharing with the N64 community your findings about the
* challenge/response pair used in 'Jet Force Gemini' and the 267
* challenge/response pairs used in 'Banjo Tooie', all stored in the
* 'pif2.dat' file of Project 64. They were instrumental to the final
* success of this endeavour.
*
* - Silicon Graphics, Inc. (SGI): For creating MIPS R4000, MIPS R4300 and
* Reality Co-Processor (RCP). You were the ultimate dream creator during
* the late 80's and early 90's. A very special word of gratitude goes to
* the two teams that during those years created RCP and MIPS R4300. They
* were technological breakthroughs back then.
*
* On a personal note, I would like to show my deepest gratitude to _Bijou_,
* for being always a source of endless hope and inspiration.
*
* -= X-Scale =- (#n64dev@EFnet)
*/
#include "n64_cic_nus_6105.h"
void n64_cic_nus_6105(int8_t chl[], int8_t rsp[], int len)
{
int i;
char key;
const char *lut;
static const char lut0[0x10] = {
0x4, 0x7, 0xA, 0x7, 0xE, 0x5, 0xE, 0x1,
0xC, 0xF, 0x8, 0xF, 0x6, 0x3, 0x6, 0x9
};
static const char lut1[0x10] = {
0x4, 0x1, 0xA, 0x7, 0xE, 0x5, 0xE, 0x1,
0xC, 0x9, 0x8, 0x5, 0x6, 0x3, 0xC, 0x9
};
for (key = 0xB, lut = lut0, i = 0; i < len; i++)
{
int mag, mod, sgn;
rsp[i] = (key + 5 * chl[i]) & 0xF;
key = lut[(int) rsp[i]];
sgn = (rsp[i] >> 3) & 0x1;
mag = ((sgn == 1) ? ~rsp[i] : rsp[i]) & 0x7;
mod = (mag % 3 == 1) ? sgn : 1 - sgn;
if (lut == lut1 && (rsp[i] == 0x1 || rsp[i] == 0x9))
mod = 1;
if (lut == lut1 && (rsp[i] == 0xB || rsp[i] == 0xE))
mod = 0;
lut = (mod == 1) ? lut1 : lut0;
}
}
| {
"pile_set_name": "Github"
} |
Three-dimensional echocardiographic characterization of patients with left ventricular noncompaction.
Despite several efforts using two-dimensional echocardiography and cardiac magnetic resonance in the diagnosis of left ventricular noncompaction (LVNC), there are no universally accepted diagnostic criteria. The aim of this study was to describe the extent of noncompacted myocardium using a new three-dimensional echocardiographic parameter. Seventeen patients with diagnoses of LVNC on the basis of two-dimensional echocardiographic and clinical criteria, 26 Olympic rowing athletes, and 49 healthy volunteers underwent three-dimensional echocardiography. By offline analysis, left ventricular volumes, mass, ejection fraction, and sphericity index were calculated. Trabeculated left ventricular volume (TLV) was calculated as the difference between left ventricular end-diastolic volume obtained including and excluding the trabeculae in the cavity contour. TLV was also normalized by left ventricular end-diastolic volume (TLV%). TLV and TLV% were significantly higher in patients with LVNC (33.7 ± 10.9 mL and 24 ± 7%) as opposed to controls (7.1 ± 2.2 mL, P < .001, and 6 ± 2%, P < .001, respectively) and athletes (8.0 ± 3.0 mL, P < .001, and 5 ± 2%, P < .001, respectively). In detail, on receiver operating characteristic curve analysis, optimal cutoff values of 15.8 mL for TLV and 12.8% for TLV% were determined for the identification of LVNC (area under the curve, 1.00; P < .001). Mild positive correlations of TLV and TLV% were found with sphericity index (r = 0.294, P = .004, and r = 0.301, P = .004, respectively), and mild negative correlations were found with ejection fraction (r = -0.454, P < .001, and r = -0.217, P = .038, respectively). Because of high spatial resolution and accuracy in volumetric quantification, three-dimensional echocardiography allows accurate measurement of the extent of noncompacted myocardium and identification of patients with LVNC. | {
"pile_set_name": "PubMed Abstracts"
} |
Frequency distribution of antimalarial drug-resistant alleles among isolates of Plasmodium falciparum in Purworejo district, Central Java Province, Indonesia.
Treatment failures to the first- and second-lines antimalarial drugs chloroquine and sulfadoxine-pyrimethamine have increased in the Purworejo district on the island of Java, Indonesia. A molecular epidemiologic study was conducted to determine the frequency distribution of mutant alleles of the genes associated with the resistance among the isolates of Plasmodium falciparum from the area. Analyses using a polymerase chain reaction and restriction fragment length polymorphism showed that nearly all of the 111 samples carried mutant alleles in genes associated with chloroquine resistance: P. falciparum multi-drug resistance 1 (pfmdrl) 86Y (92%), 1042D (4.5%), and P. falciparum chloroquine resistance transporter (pfcrt) 76T (99.1%). Mutant alleles of the in the dihydrofolate reductase (dhfr) gene were also high (84.7%), either as 108N and 108T or paired with 59R, and 16V, respectively. Mutant alleles in the dihydropteroate synthase gene were the least common, either as a single 437G mutation (35.3%) or paired with 540E (26.5%). These results are consistent with the antimalarial drug resistance situation in the area and emphasize the need for a proper treatment strategy. | {
"pile_set_name": "PubMed Abstracts"
} |
---
address: 'Mathematisches Institut, Universität Freiburg, D-79102 Freiburg'
author:
- Bernd Siebert
date: 'February 5, 1997; revision of August 14, 2005'
title: 'Virtual fundamental classes, global normal cones and Fulton’s canonical classes'
---
[**Introduction**]{}
This note, written in January 1997, grew out of an attempt to understand references [@behrend], [@behrendfantechi] and [@litian]. In these papers two related but different methods are presented for the construction of a certain Chow class on moduli spaces of stable (parametrized) curves in a projective manifold $V$, called virtual fundamental class. This class replaces the usual fundamental class of these spaces in the definition of basic enumerative invariants of $V$ involving curves, called Gromov-Witten (GW-) invariants. They are invariant under smooth deformations of $V$.
Both approaches are based on a globalization of the concept of normal cones of germs of the space under study inside some modelling space, that is $C_{U|M}$ for $U\subset X$ open with $\iota: U
\hookrightarrow M$ and $M$ smooth over $k$. The essential idea of using bundles of cones inside a vector bundle for globalizing virtual fundamental classes is due to Li and Tian. The data needed to glue differs however somewhat in the two constructions.
A proper understanding of the relationship between the two approaches seemed necessary for finding the natural framework for comparison of algebraic virtual fundamental classes with the author’s definition in [@si1] of virtual fundamental classes in the symplectic context [@si3].
In a first step Behrend and Fantechi use a generalization of the concept of scheme, called Artin stacks, to make sense of the quotient $C_{U|M}/T_M|_U$. These quotients being unique up to canonical isomorphism they glue to an Artin (cone) stack ${{{\mathcal C}}}_X$ intrinsically associated to any $X$. In a second step they need a morphism $\ph^\bullet: [{{\mathcal F}}^{-1} \rightarrow {{\mathcal F}}^0] \rightarrow
{{\mathcal L}}_X^\bullet$ (in the derived category) from a two-term complex of locally free sheaves to the cotangent complex, inducing an isomorphism in $H^0$ and an epimorphism in $H^{-1}$, to cook up an ordinary cone $C(\ph^\bullet)\subset F_1$, $F_1$ the vector bundle associated to ${{\mathcal F}}^{-1}$. Intersection with the zero section finally produces the virtual fundamental class. So a priori the latter depends on the choice of $\ph^\bullet$.
This is a very natural and mature approach, that clearly separates the globalization process of the normal cone from the construction of the virtual fundamental class. A possible disadvantage is that in dealing with Artin stacks some of the necessary verifications become rather technical, non-geometric in nature.
Li and Tian circumvent the morphism to the cotangent complex by introducing the notion of “perfect tangent obstruction complex”. This is a morphism ${{\mathcal F}}^{-1}\rightarrow{{\mathcal F}}^0$ of locally free sheaves on $X$ with kernel and cokernel being tangent and obstruction spaces for morphisms to $X$, compatible with base change. Using relative, formal, “Kuranishi families” as an intermediate object they construct a well-defined cone $C\subset F_1$. Another, less important difference to [@behrendfantechi] is the use of an absolute obstruction theory instead of one relative to the space of pre-stable curves.
In a previous version of [@litian] the slightly stronger claim was made that already a presentation ${{\mathcal F}}^{-1}\rightarrow{{\mathcal F}}^0$ of $\Omega_X$ should suffice to construct the cone. In trying to understand this statement I was lead to the problem of reformulating [@behrendfantechi] from the point of view of gluing local cones. Since in the latter reference Artin stacks are used only as book-keeping device rather than as actual spaces it should not come as surprise that one can get along without them (this has already been indicated in op.cit.). Contrary to what I expected, things can be formulated in a rather elegant but direct way via some *yoga of cones bundles*. This part of the paper (Sections 2 and 3) is just a down-to-earth reformulation of (parts of) Sections 2–4 of [@behrendfantechi]. Section 1 presents the necessary notations concerning cones and linear spaces, the latter being a convenient way of looking at coherent sheaves for our purposes.
In Section 4 we establish a closed formula for virtual fundamental classes involving only the scheme-theoretic structure of $X$ via Fulton’s canonical class (Definition \[fultons\_class\]) and the Chern class of the virtual bundle $F_0-F_1$, $F_i$ the vector bundle associated to ${{\mathcal F}}^{-i}$ (Theorem \[closed\_formula\]). This formula was actually found by the author in summer 1995 while searching for a purely algebraic definition of GW-invariants. It should be useful for computations, see the author’s recent little survey [@si2].
A few remarks on GW-theory are in order. First, today I consider the yoga of cone bundles in Sections 2 and 3 as one ingredient for the most economic path to algebraic Gromov-Witten invariants. The other ingredients are going over to Deligne-Mumford stacks, and replacing the morphism to the cotangent complex by an obstruction theory. If the latter is defined similar to [@artin], 2.6, rather than in [@litian], one can show [@si4] that it is locally nothing but a morphism to the cotangent complex as in [@behrendfantechi]. Hence the yoga of cone bundles applies to produce the virtual fundamental class.
Second, I would like to illustrate the perspective of the content of Section 4 by the following formula for virtual fundamental classes in Gromov-Witten theory.
Let $V$ be a projective variety, smooth over a field $K$ of characteristic $0$, and $R\in A_1(V)$, the first Chow group. If $g=0$ or ${{\mathcal C}}:= {{\mathcal C}}_{R,g,k}(V)$, the moduli space of stable curves $(C,{\bf x},\ph:C\rightarrow V)$ in $V$ of genus $g$ with $k$ marked points ${\bf x}= (x_1,\ldots,x_k)$ and $\ph_*[C]=R$, is embeddable into a space smooth over ${\mathfrak{M}}_{g,k}$, then the virtual fundamental class relevant for GW-invariants is $$[\![{{\mathcal C}}]\!]\ =\ \Big\{c(\ind_{R,g,k}^V)^{-1}\cap
c_F({{\mathcal C}}/{\mathfrak{M}}_{g,k})\Big\}_{d(V,R,g,k)}\, .$$ Here $\{\,.\,\}_d$ denotes the $d$-dimensional part of a cycle, ${\mathfrak{M}}_{g,k}$ is the Artin stack of $k$-pointed pre-stable curves of genus $g$, $d(V,R,g,k)= c_1(V)\cdot R +(1-g) \dim V+ 3g-3$ is the expected dimension, and $c_F({{\mathcal C}}/ {\mathfrak{M}}_{g,k})$ is Fulton’s canonical class for ${{\mathcal C}}$ relative ${\mathfrak{M}}_{g,k}$.
Here $\ind_{R,g,k}^V=F_0-F_1$ is the virtual vector bundle associated to the partial resolution $\varphi^\bullet: [{{\mathcal F}}^{-1}\to
{{\mathcal F}}^0]\to {{\mathcal L}}^\bullet_{{{\mathcal C}}/ {\mathfrak{M}}_{g,k}}$ mentioned in the introduction. It represents the (domain of the) perfect relative obstruction theory $(R\pi_*(f^T_V))^\vee$ of Behrend [@behrend] in $K^0({{\mathcal C}})$.
A note on categories: To keep things simple we work here in the category of schemes of finite type over a field $k$, not necessarily algebraically closed or of characteristic 0. The extension to other base schemes is straightforward. For the purpose of GW-theory one also has to replace schemes by (generalizations of) orbifolds, that is Deligne-Mumford stacks in the algebraic category or analytic orbispaces in an analytic context. Again, our results can be easily adapted to these categories.
For GW-theory this is still not sufficient, because ${\mathfrak{M}}_{g,k}$ is an Artin stack rather than Deligne-Mumford. One can nevertheless give a construction of the relevant cone without ever really using Artin stacks. For instance, Fulton’s canonical class relative ${\mathfrak{M}}_{g,k}$ has the following simple definition: Embed ${{\mathcal C}}$ into a smooth Deligne-Mumford $k$-stack $N$. In the important case $g=0$ one could take ${{\mathcal C}}_{\iota_*R,g,k}
(\pr^N)$, if $\iota: V\hookrightarrow \pr^N$ is a closed embedding. Let $q:{{\mathcal U}}\rightarrow {{\mathcal C}}$ be the universal curve. Then the pull-back of the (virtual) tangent bundle of ${\mathfrak{M}}_{g,k}$ is $$T_{\mathfrak{M}} := {\mathcal{E} \!\textrm{\textit{xt}}}^1_q (\omega_{{{\mathcal U}}/{{\mathcal C}}},{{\mathcal O}}_{{\mathcal U}})-
{\mathcal{E} \!\textrm{\textit{xt}}}^0_q (\omega_{{{\mathcal U}}/{{\mathcal C}}},{{\mathcal O}}_{{\mathcal U}})\,,$$ as an element of $K^0({{\mathcal C}})$ (the ${\mathcal{E} \!\textrm{\textit{xt}}}^i_q$ are the derived functors of $q_*\circ{\mathcal{H} \!\textrm{\textit{om}}}$) and $$\begin{aligned}
c_F({{\mathcal C}}/ {\mathfrak{M}}_{g,k})&=&
\Big(c(T_{\mathfrak{M}})^{-1} \cup c(T_N)\Big) \cap s(C_{{{\mathcal C}}/N})\\
&=& c(T_{\mathfrak{M}})^{-1}\cap c_F({{\mathcal C}})\,.\end{aligned}$$ Here all sheaves and classes have to be understood in the sense of Deligne-Mumford stacks. With these remarks understood the theorem is a special case of Theorem \[closed\_formula\].
In preparing this paper discussions with H. Flenner and S.Schröer have been helpful. I am grateful to the referee for a very attentive reading of the manuscript and several competent suggestions.
Cones
=====
Linear spaces
-------------
For any algebraic $k$-scheme $X$, $\Aff^1_X=X\times\Aff^1_k$ has the structure of a [*ring over*]{} $X$: There are morphisms $$\alpha:\Aff^1_X\times_X\Aff^1_X\longrightarrow\Aff^1_X\, ,\quad
\iota:\Aff^1_X\longrightarrow\Aff^1_X\, ,\quad
\mu:\Aff^1_X\times_X\Aff^1_X\longrightarrow\Aff^1_X\, ,$$ and sections $n,e:X\rightarrow\Aff^1_X$ fulfilling the usual commutative ring axioms with $\alpha$ as addition, $\iota$ as additive inverse, $\mu$ as multiplication and $n$, $e$ as neutral elements for $\alpha$, $\mu$.
A [*linear space over $X$*]{} is an $\Aff^1_X$-module of finite type over $X$, that is, an affine morphism $\pi:L\rightarrow X$ of finite type together with morphisms $$a:L\times_X L\longrightarrow L,\quad m:\Aff^1_X\times_X L
\longrightarrow L$$ and a zero section $z:X\rightarrow L$, fulfilling the usual module axioms relative $X$, that is $m\circ(\mu\times\Id_L) =
m\circ(\Id_{\Aff^1_X}\times m)$ as maps from $\Aff^1_X \times_X
\Aff^1_X\times_X L$ to $L$ etc.[^1] By abuse of notation we just write $L$ for the tuple $(\pi,a,m)$. In the sequel we will restrict to linear spaces that are [*representable*]{}, that is, which locally are closed subspaces of vector bundles with induced linear structure. With the obvious notion of homomorphism of linear spaces over $X$ we get the category ${{\mathop{\rm Lin}}}(X)$ of representable linear spaces over $X$.
There is an anti-equivalence of categories $${{\mathop{\rm Lin}}}(X)\longrightarrow{{\mathop{\rm Coh}}}(X)$$ to the category of coherent ${{\mathcal O}}_X$-modules [@EGA-II],§1.7: On objects this associates to $L\in {{\mathop{\rm Lin}}}(X)$ the sheaf $\Hom_{{{\mathop{\rm Lin}}}(X)}(L,\Aff^1_X)$; in the other direction, ${{\mathcal F}}\in{{\mathop{\rm Coh}}}(X)$ corresponds to $$L({{\mathcal F}})\ :=\ \Spec_{{{\mathcal O}}_X}S^\bullet{{\mathcal F}}\, ,$$ where $S^\bullet{{\mathcal F}}$ is the symmetric algebra over the ${{\mathcal O}}_X$-module ${{\mathcal F}}$. For example, the addition operation $a:
L({{\mathcal F}})\times_X L({{\mathcal F}})\to L({{\mathcal F}})$ comes from the diagonal morphism ${{\mathcal F}}\to {{\mathcal F}}\oplus {{\mathcal F}}$, $f\mapsto (f,f)$ by application of the functor $L=\Spec_{{{\mathcal O}}_X}\circ S^\bullet$. Note that a vector bundle $E$ corresponds to the locally free sheaf ${{\mathcal O}}(E^\vee)$, $E^\vee$ the [*dual*]{} bundle.
Representable linear spaces are thus just another way to look at coherent sheaves. We will jump freely between both descriptions and use whichever seems more appropriate in a particular context. Note also that ${{\mathop{\rm Lin}}}(X)$ is an abelian category, so it makes sense to talk about monomorphisms, epimorphisms and exact sequences. A monomorphism $\Phi:E\rightarrow F$ of linear spaces corresponds to an epimorphism $\ph:{{\mathcal F}}\rightarrow{{\mathcal E}}$ of sheaves and is thus a closed embedding of schemes. An epimorphism $\psi:F\rightarrow G$ of linear spaces, however, need not be a surjection of schemes (consider the inclusion $\psi:{{\mathcal I}}\hookrightarrow{{\mathcal O}}_X$ for any nontrivial ideal sheaf ${{\mathcal I}}$).
Cones
-----
A [*cone*]{} $C$ over $X$ is a scheme of the form $\Spec_{{{\mathcal O}}_X}
S^\bullet$ where $S^\bullet= \oplus_{d\ge0}S^d$ is a graded ${{\mathcal O}}_X$-module with $S^0={{\mathcal O}}_X$ and $S^\bullet$ generated by $S^1\in{{\mathop{\rm Coh}}}(X)$. $S^\bullet$ as graded algebra is not in general determined up to isomorphism by the scheme $C$ over $X$. For the grading one needs to distinguish the generating submodule $S^1={{\mathcal F}}$, or, equivalently, a closed embedding $C\hookrightarrow
L({{\mathcal F}})$ into a linear space. Such datum could be called [*polarization*]{} of $C$. We will only deal with polarized cones in the sequel.
\[normal\_cone\]*If $X$ is a closed subscheme of an algebraic $k$-scheme $M$ with ideal sheaf ${{\mathcal I}}$ then the cone $$C_{X|M}=\Spec_{{{\mathcal O}}_X}\big(\oplus_{d\ge0}{{\mathcal I}}^d/{{\mathcal I}}^{d+1}\big)$$ over $X$ is called [*normal cone*]{} to $X$ in $M$. $C_{X|M}$ is naturally embedded into the [*normal space*]{} $N_{X|M}=L({{\mathcal I}}/{{\mathcal I}}^2)$ of $X$ in $M$ (to avoid confusion with $({{\mathcal I}}/{{\mathcal I}}^2)^\vee$, I would rather not call $N_{X|M}$ normal sheaf as in [@behrendfantechi]).*
If $C$, $C'$ are cones over $X$, then so is $C\oplus C':=C\times_X C'$.
To a polarized cone $C=\Spec S^\bullet$ is associated a Chow class on $X$, its Segre class $$s(C)\ :=\ \sum_{r\ge0}p_*\big(\xi^r\cap[\pr(C)]\big),$$ where $p:\pr(C) :=\Proj S^\bullet\rightarrow X$ is the projection and $\xi = c_1({{\mathcal O}}_{\pr(C)}(1))$.
We propose the following formulation of the concept of exact sequence of cones [@fulton Expl.4.1.6].
Let $$\mbox{\hspace*{4cm}}
0\longrightarrow E\stackrel{\Phi}{\longrightarrow}F
\stackrel{\Psi}{\longrightarrow}Q\longrightarrow 0
\mbox{\hspace*{4cm}}(*)$$ be an exact sequence of linear spaces. Let $C\subset Q$ be a cone and set $\tilde C:=\Psi^{-1}(C)$. Then $(*)$ restricts to $$0\longrightarrow E\longrightarrow\tilde C\longrightarrow C
\longrightarrow 0 \quad.$$ Sequences of cones of this form will be called [*exact*]{}.
*Exact sequences of cones might not be very useful unless $(*)$ splits locally. In this case $\tilde C$ is locally of the form $C\oplus E$, and as in [@fulton Expl.4.1.6] one can show $s(\tilde C)=s(C\oplus
E)$. In the non-split case a convenient way to relate the Segre classes of $\tilde C$ and $C$ seem to be unknown.*
But note that if $E$ is a vector bundle $(*)$ always splits locally, and so we retrieve the definition of exact sequences of cones as in [@fulton].
For an exact sequence of cones as in the definition $\tilde C$ is preserved by the additive action of $E$ on $F$. In other words, $\tilde C$ wears the structure of an $E$-module. More generally, if $\Phi:E\rightarrow F$ is a homomorphism of linear spaces and $C\subset F$ is a cone then $C$ is called [*$E$-cone*]{} if $C$ is an $E$-module via $\Phi$, that is if $C$ is preserved by the additive action of $E$ on $F$ induced by $\Phi$.
\[C\_Z|X\] In the situation of Example \[normal\_cone\] $C_{X|M}$ is a $T_M|_X$-cone via the natural homomorphism $\Phi:T_M|_X
\rightarrow N_{X|M}$. On the sheaf level this action of $T_M|_X$ is $$\begin{aligned}
\bigoplus_d {{\mathcal I}}^d/{{\mathcal I}}^{d+1}&\longrightarrow&
S^\bullet \Omega_M|_X \otimes \bigoplus_d {{\mathcal I}}^d/{{\mathcal I}}^{d+1},\end{aligned}$$ where for $f_i\in {{\mathcal I}}/{{\mathcal I}}^2$ the image of $f_1\cdot\ldots\cdot f_d$ in the direct summand $S^e \Omega_M|_X \otimes {{\mathcal I}}^{d-e}/{{\mathcal I}}^{d-e+1}$ of the target is the sum over all partitions $\{i_1,\dots,i_e\}$, $\{j_1,\dots,j_{d-e}\}$ of $\{1,\dots,n\}$ of terms $$\begin{aligned}
\textrm{d}f_{i_1}\cdot \ldots\cdot \textrm{d}f_{i_e}\otimes
f_{j_1}\cdot\ldots\cdot f_{j_{d-e}}.\end{aligned}$$
There are many examples of morphisms of linear spaces $E\rightarrow F$ and $E$-cones $C\subset F$ that do not descend to the quotient $Q=E/F$, for instance the examples in Remark \[coh\_cones\],3 and in Remark \[C\_X|M\_descend?\]. However, there is one important class of morphisms where it is always possible, namely for locally split monomorphisms. We first treat the split case:
\[C\_in\_EplusF\] Let ${{\mathcal E}}$, ${{\mathcal F}}\in{{\mathop{\rm Coh}}}(X)$ and $E=L({{\mathcal E}})$, $F=L({{\mathcal F}})$ the corresponding linear spaces over $X$ and $C\subset E\oplus F$ an $E$-invariant closed subscheme with respect to the action of $E$ on the first summand.
Then $C$ is of the form $E\oplus\bar C$ for some uniquely determined closed subscheme $\bar C\subset F$.
[[*Proof.* ]{}]{}The statement is local in $X$, so we may assume $X=\Spec A$, $E=\Spec A[{\bf X}]/\langle{\bf e}\rangle$, $F=\Spec A[{\bf Y}]/
\langle{\bf f}\rangle$ with ${\bf X}=(X_1,\ldots X_r)$, ${\bf Y}=
(Y_1,\ldots Y_s)$ and ${\bf e}=(e_1,\ldots,e_k)$, ${\bf
f}=(f_1,\ldots,f_l)$ tuples of linear forms with coefficients in $A$, $\langle{\bf e}\rangle$, $\langle{\bf f}\rangle$ the ideals generated by their entries. Then $C=\Spec A[{\bf X},{\bf Y}]/I$ with $I$ an ideal containing $\langle{\bf e}\rangle+\langle{\bf f}\rangle$.
The only possible candidate for $\bar C$ is the intersection of $C$ with $0\oplus F$, that is $\bar C=\Spec A[{\bf X},{\bf Y}]/(I+\langle{\bf
X}\rangle)=\Spec A[{\bf Y}]/\bar I$, with $\bar I=\{f(0,{\bf Y})\mid f({\bf
X},{\bf Y})\in I\}$. We have to show that $I=\langle\,\bar
I\,\rangle+\langle{\bf e}\rangle$.
$C$ to be $E$-invariant means that for any $\displaystyle f({\bf X},
{\bf Y})=\sum_{M,N}a_{MN}{\bf X}^M{\bf Y}^N\in I$ $$\mbox{\hspace*{2cm}}
f({\bf X}+{\bf X'},{\bf Y})\ =\ \sum_{M,N}a_{MN}({\bf X}+{\bf X'})^M
{\bf Y}^N \ \in\ \langle\,I\,\rangle+\langle\ph({\bf e})
\rangle\mbox{\hspace*{2cm}}(*)$$ holds in $A[{\bf X},{\bf X'},{\bf Y}]$, where $\ph:A[{\bf X}] \rightarrow
A[{\bf X'}]$, $X_\mu\mapsto X_\mu'$. Modulo $\langle{\bf X}\rangle$ this says $$f({\bf X'},{\bf Y})\in\langle\,\bar I\,\rangle+\langle\ph({\bf e})\rangle$$ in $A[{\bf X'},{\bf Y}]$. Replacing $\bf X'$ by $\bf X$ we thus get $I\subset\langle\,\bar I\,\rangle+\langle{\bf e}\rangle$.
For the other direction we look at $(*)$ modulo ${\bf X}+{\bf X'}$ to conclude $$f(0,{\bf Y})\ =\ \sum_N a_{0N}{\bf Y}^N
\ \in\ I+{\bf e}\ =\ I$$ for any $f\in I$, that is $\bar I\subset I$.
\[quot\_cone\] Let $$\mbox{\hspace*{5cm}}0\longrightarrow F\longrightarrow E
\stackrel{q}{\longrightarrow}Q
\mbox{\hspace*{5cm}}(*)$$ be an exact sequence of linear spaces with $F$ a vector bundle, and let $C\subset E$ be an $F$-cone.
Then there exists a unique cone $\bar C\subset Q$ such that $(*)$ induces an exact sequence of cones $$0\longrightarrow F\longrightarrow C\longrightarrow\bar C
\longrightarrow0\, .$$ In particular, $C$ descends to $Q$: $C=q^{-1}(\bar C)$.
[[*Proof.* ]{}]{}By replacing $Q$ by the closed subspace $E/F\subset Q$ we may assume $q$ to be an epimorphism. Then, since $F$ is a vector bundle, locally $(*)$ splits and we may apply the previous lemma to construct $\bar C\subset Q$.
In other words, the proposition says that $\bar C$ is the scheme-theoretic quotient of $C$ by the free action of $F$. This is a convenient way to think about $\bar C$.
Going up and down for $E$-cones
===============================
In this section we investigate the behavior of $E$-cones under morphisms of two-term complexes, that is commutative squares, in ${{\mathop{\rm Lin}}}(X)$. If $\Phi_\bullet=(\Phi_0,\Phi_1):
F_\bullet=(F_0\rightarrow F_1)\rightarrow (E_0\rightarrow E_1)$ is such a morphism the corresponding morphism of coherent sheaves will be written $\ph^\bullet=(\ph^{-1},\ph^0):{{\mathcal E}}^\bullet
=({{\mathcal E}}^{-1}\rightarrow{{\mathcal E}}^0)\rightarrow{{\mathcal F}}^\bullet=
({{\mathcal F}}^{-1}\rightarrow{{\mathcal F}}^0)$. Then $\Phi_i=L(\ph^{-i})$, $E_i=L({{\mathcal E}}^{-i})$, $F_i=L({{\mathcal F}}^{-i})$ for $i=0,1$.
Going up
--------
Let $\Phi_\bullet:F_\bullet\rightarrow E_\bullet$ be a commutative square in ${{\mathop{\rm Lin}}}(X)$, and $C\hookrightarrow E_1$ an $E_0$-cone. Then $\Phi_1^{-1}(C)\hookrightarrow F_1$ is an $F_0$-cone.
[[*Proof.* ]{}]{}Consider the diagram $$\begin{array}{ccc}
F_0\oplus F_1&\stackrel{\alpha}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&F_1\\[10pt]
{\makebox[0cm]{${\scriptstyle\Phi_0\oplus\Phi_1\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\Phi_0\oplus\Phi_1\ }$}}&&{\makebox[0cm]{$\phantom{\ \scriptstyle\Phi_1}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle\Phi_1}$}}\\[10pt]
E_0\oplus E_1&\stackrel{\alpha'}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&E_1
\end{array}$$ with horizontal arrows the morphisms defining the $F_0$- and $E_0$-module structures on $F_1$ and $E_1$ respectively. By hypothesis $E_0\oplus C$ is a closed subscheme of $(\alpha')^{-1}(C)$. Thus $F_0\oplus\Phi_1^{-1}(C)
=(\Phi_0\oplus\Phi_1)^{-1} (E_0\oplus C)$ is a closed subscheme of $\alpha^{-1} (\Phi_1^{-1}(C))$.
By this lemma we are able to make the following definition.
(going up) Let $\Phi_\bullet:F_\bullet\rightarrow E_\bullet$ be a commutative square in ${{\mathop{\rm Lin}}}(X)$ and $C\subset E_1$ an $E_0$-cone. Then the $F_0$-cone $$\Phi_\bullet^!(C):=\Phi_1^{-1}(C)$$ in $F_1$ is called [*pull-back*]{} of $C$ under $\Phi_\bullet$.
The pull-back depends only on the homotopy class of $\ph^\bullet$ (or $\Phi_\bullet$).
Let $\ph^\bullet, \psi^\bullet:{{\mathcal E}}^\bullet=[{{\mathcal E}}^{-1} \stackrel{d}{\rightarrow}
{{\mathcal E}}^0] \rightarrow {{\mathcal F}}^\bullet$ be homotopic commutative squares in ${{\mathop{\rm Coh}}}(X)$. Then for any $E_0$-cone $C\subset E_1$ $$\Phi_\bullet^!(C)\ =\ \Psi_\bullet^!(C)\, .$$
[[*Proof.* ]{}]{}Let $k:{{\mathcal E}}^0\to {{\mathcal F}}^{-1}$ be a homotopy: $\psi^{-1} =
\ph^{-1}+k\circ d$, $\psi^0=\ph^0+ d\circ k$. Writing $K=L(k)$ and $\alpha:E_0\oplus E_1\rightarrow E_1$ for the structure map, $\Psi_1$ may be decomposed into $$F_1\stackrel{(K,\Phi_1)}{\longrightarrow}E_0\oplus E_1
\stackrel{\alpha}{\longrightarrow}E_1\, .$$ Since $E_0\oplus C\subset\alpha^{-1}(C)$, $(K,\Phi_1)^{-1}
(E_0\oplus C) =\Phi_1^{-1}(C)$ is a closed subscheme of $\Psi_1^{-1}(C)$. But the claim is symmetric in $\Phi_\bullet$, $\Psi_\bullet$, hence $\Phi_1^{-1}(C)=\Psi_1^{-1}(C)$.
The next result about functoriality of going up follows directly from the definition.
\[gu\_functorial\] Let $\Phi_\bullet:E_\bullet\rightarrow F_\bullet$, $\Psi_\bullet:
F_\bullet\rightarrow G_\bullet$ be commutative squares of linear spaces and $C\subset G_1$ a $G_0$-cone. Then $$(\Psi_\bullet\circ\Phi_\bullet)^!(C)\ =\ \Phi_\bullet^!\circ
\Psi_\bullet^!(C)\, .$$
Going down
----------
Going down, or push-forward, of $F_0$-cones in $F_1$ to $E_1$ is a little more subtle. The central tool will be Proposition \[quot\_cone\]. To make this proposition applicable we need a little lemma.
\[ass\_exact\_sequence\] Let $\ph^\bullet:({{\mathcal E}}^{-1}\stackrel{d}{\rightarrow}{{\mathcal E}}^0)
\rightarrow ({{\mathcal F}}^{-1}\stackrel{d'}{\rightarrow}{{\mathcal F}}^0)$ be a commutative square in ${{\mathop{\rm Coh}}}(X)$. Then the complex $$0\longrightarrow{{\mathcal E}}^{-1}\stackrel{(d,\ph^{-1})}{\longrightarrow}
{{\mathcal E}}^0\oplus{{\mathcal F}}^{-1}\stackrel{\ph^0\circ\prj_1
-d'\circ\prj_2}{\longrightarrow}{{\mathcal F}}^0\longrightarrow0$$ is exact at
------ --------------------------------------------- -----------------------------------------------------------------------------
i) ${{\mathcal F}}^0$ iff $H^0(\ph^\bullet)$ is surjective
ii) ${{\mathcal E}}^0\oplus{{\mathcal F}}^{-1}$ iff $H^0(\ph^\bullet)$ is injective and $H^{-1}(\ph^\bullet)$ is surjective
iii) ${{\mathcal E}}^{-1}$ iff $H^{-1}(\ph^\bullet)$ is injective.
------ --------------------------------------------- -----------------------------------------------------------------------------
[[*Proof.* ]{}]{}Chase the diagram $$\begin{array}{rcccccccccccl}
\phantom{.}&
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal K}}&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal E}}^{-1}
&\stackrel{d}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&{{\mathcal E}}^0
&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal Q}}&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0\\[8pt]
&&&{\makebox[0cm]{${\scriptstyleH^{-1}(\ph^\bullet)\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyleH^{-1}(\ph^\bullet)\ }$}}&&{\makebox[0cm]{${\scriptstyle\ph^{-1}\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ph^{-1}\ }$}}&&
{\makebox[0cm]{$\phantom{\ \scriptstyle\ph^0}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle\ph^0}$}}&&{\makebox[0cm]{$\phantom{\ \scriptstyleH^0(\ph^\bullet)}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyleH^0(\ph^\bullet)}$}}\\[12pt]
&0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal K}}'&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal F}}^{-1}
&\stackrel{d'}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&{{\mathcal F}}^0
&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{{\mathcal Q}}'&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0&.
\end{array}$$
If $\ph^\bullet$ is a quasi-isomorphism we thus get exactness of the stated complex. And $\ph^\bullet$, viewed as a commutative square, is cartesian (${{\mathcal E}}^{-1}={{\mathcal E}}^0\oplus_{{{\mathcal F}}^0}{{\mathcal F}}^{-1}$) iff $H^0(\ph^\bullet)$ is injective and $H^{-1}(\ph^\bullet)$ is an isomorphism, and it is cocartesian (${{\mathcal F}}^0=({{\mathcal E}}^0\oplus
{{\mathcal F}}^{-1})/ {{\mathcal E}}^{-1}$) iff $H^0(\ph^\bullet)$ is an isomorphism and $H^{-1}(\ph^\bullet)$ is surjective. Assume now that $F_0$ is a vector bundle and that $\Phi_\bullet:[F_0
\stackrel{D'}{\rightarrow}F_1]\rightarrow[E_0
\stackrel{D}{\rightarrow}E_1]$ induces an isomorphism on $H^0$ and a closed embedding of linear spaces on $H^1$. If these conditions are satisfied we say that [*going down is applicable to $\Phi_\bullet$*]{}. Then $$0\longrightarrow F_0\stackrel{(\Phi_0,-D')}{\longrightarrow}
E_0\oplus F_1 \stackrel{q}{\longrightarrow}E_1$$ is exact (Lemma \[ass\_exact\_sequence\], $q=D\circ\prj_1
+\Phi_1\circ\prj_2$) and we may apply Proposition \[quot\_cone\].
\[going\_down\] (going down) Let $\Phi_\bullet:F_\bullet\rightarrow E_\bullet$ be a commutative square in ${{\mathop{\rm Lin}}}(X)$, to which going down is applicable (see above), and let $C\subset F_1$ be an $F_0$-cone. The unique cone $\bar C\subset\img q\subset E_1$ with $q^{-1}(\bar
C)=E_0\oplus C$, which exists by Proposition \[quot\_cone\], is called [*push-forward*]{} of $C$ by $\Phi_\bullet$, denoted $(\Phi_\bullet)_!(C)$.
Note that $(\Phi_\bullet)_!(C)$ is actually an $E_0$-cone because $E_0\oplus C$ is one. And by Proposition \[quot\_cone\]:
\[ex\_seq\_cones\] If going down is applicable to $\Phi_\bullet:F_\bullet\rightarrow
E_\bullet$, and $C\subset F_1$ is an $F_0$-cone, there is an exact sequence of cones $$0\longrightarrow F_0\longrightarrow E_0\oplus C\longrightarrow
(\Phi_\bullet)_!(C)\longrightarrow0\, .$$
Local freeness of $F_0$ (or local splittability of the relevant exact sequence of linear spaces) seems to be indispensable, since otherwise $E_0\oplus C$ need not descend to $E_1$. See Remark \[coh\_cones\],3 for a related example.
As with going up, going down depends only on the homotopy class of $\Phi_\bullet$.
Let $\Phi_\bullet$, $\Psi_\bullet:F_\bullet\rightarrow E_\bullet$ be homotopic morphisms of commutative squares in ${{\mathop{\rm Lin}}}(X)$ and $C\subset F_1$ an $F_0$-cone. If going down is applicable to $\Phi_\bullet$ (or, equivalently, to $\Psi_\bullet$) then $$(\Phi_\bullet)_!(C)\ =\ (\Psi_\bullet)_!(C)\, .$$
[[*Proof.* ]{}]{}Let $K:F_1\rightarrow E_0$ be a homotopy between $\Phi_\bullet$ and $\Psi_\bullet$, that is $\Psi_0=\Phi_0+K\circ D'$, $\Psi_1=
\Phi_1+D\circ K$ ($D:E_0\rightarrow E_1$, $D':F_0 \rightarrow F_1$ the differentials). Then the following diagram $$\begin{array}{ccccccc}
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&F_0&\stackrel{\makebox[0pt]{$\scriptstyle
(\Phi_0,-D')$}}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&E_0\oplus F_1
&\stackrel{q_\Phi}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&E_1\\[10pt]
&&{\makebox[0cm]{${\scriptstyle\Id\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\Id\ }$}}&&{\makebox[0cm]{${\scriptstyle\chi\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\chi\ }$}}&&{\makebox[0cm]{${\scriptstyle\Id\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\Id\ }$}}\\[10pt]
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&F_0&\stackrel{\makebox[0pt]{$\scriptstyle
(\Psi_0,-D')$}}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&E_0\oplus F_1
&\stackrel{q_\Psi}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&E_1
\end{array}$$ with $\chi=(\prj_1-K\circ\prj_2,\prj_2)$, $q_\Phi=D\circ\prj_1+
\Phi_1\circ\prj_2$ and $q_\Psi=D\circ\prj_1+\Psi_1\circ\prj_2$, is commutative. Now $\chi^{-1}(E_0\oplus C) =E_0\oplus C$ and the conclusion follows from the definition of going down.
We observe also that since $q^{-1}(\bar C)=E_0\oplus C\subset
E_0\oplus F_1$ and $q|_{0\oplus F_1}=\Phi_1$, $\Phi_1^{-1}(\bar C)=C$. In other words:
\[left\_inverse\] Whenever going down is applicable to $\Phi_\bullet:F_\bullet
\rightarrow E_\bullet$ then $\Phi_\bullet^!$ is a left inverse to $(\Phi_\bullet)_!$, that is $$\Phi_\bullet^!(\Phi_\bullet)_!(C)\ =\ C$$ for any $F_0$-cone $C\subset F_1$.
Note that $\Phi_\bullet^!$ is generally not right-inverse to $(\Phi_\bullet)_!$. For example consider $\Phi_\bullet=(\Id,\iota):
(F_0\to F_1)\to (F_0\to F_1\oplus N)$ for any linear space $N$ over $X$, $\iota: F_1\to F_1\oplus N$ the inclusion of the first factor and $F_0$ acting trivially on $N$. Then for an $F_0$-cone of the form $C\oplus N$ it holds $(\Phi_\bullet)_!\Phi_\bullet^!(C\oplus N) =
C\oplus 0$. Compare however Proposition \[1-1-corr\].
Going down is functorial:
Let $\Psi_\bullet:G_\bullet\rightarrow F_\bullet$, $\Phi_\bullet:
F_\bullet\rightarrow E_\bullet$ be commutative squares of linear spaces to which going down is applicable, and let $C\subset G_1$ be a $G_0$-cone. Then $$(\Phi_\bullet\circ\Psi_\bullet)_!(C)
\ =\ (\Phi_\bullet)_! (\Psi_\bullet)_!(C)\, .$$
[[*Proof.* ]{}]{}Consider the following diagram of linear spaces and cones: $$\begin{array}{ccccccccc}
&&0&&0\\[8pt]
&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}\\[14pt]
&&G_0&\stackrel{\Id_{G_0}}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&G_0\\[5pt]
&&{\makebox[0cm]{${\scriptstyle
\bigg(\!\!\!\begin{array}{c}
\scriptstyle\Psi_0\\ \scriptstyle\Id_{G_0}
\end{array}\!\!\!\bigg)\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle
\bigg(\!\!\!\begin{array}{c}
\scriptstyle\Psi_0\\ \scriptstyle\Id_{G_0}
\end{array}\!\!\!\bigg)\ }$}}
&&{\makebox[0cm]{$\phantom{\ \scriptstyle
\left(\!\!\!\begin{array}{c}
\scriptstyle\Psi_0\\ \scriptstyle 0\\ \scriptstyle -D_G
\end{array}\!\!\!\right)}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle
\left(\!\!\!\begin{array}{c}
\scriptstyle\Psi_0\\ \scriptstyle 0\\ \scriptstyle -D_G
\end{array}\!\!\!\right)}$}}\\[-30pt]
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&F_0\oplus G_0
&\stackrel{\left(\!\!\!\begin{array}{cc}
\scriptstyle\Id_{F_0}& \scriptstyle 0\\
\scriptstyle -\Phi_0& \scriptstyle \Phi_0\circ\Psi_0\\
\scriptstyle 0& \scriptstyle -D_G
\end{array}\!\!\!\right)}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}&F_0\oplus E_0\oplus C
&\stackrel{(\Phi_1\circ D_F, D_E, \Phi_1\circ\Psi_1)}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}
&(\Phi_\bullet\circ \Psi_\bullet)_! C&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0\\[8pt]
&&{\makebox[0cm]{${\scriptstyle(-\Id_{F_0},\Psi_0)\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle(-\Id_{F_0},\Psi_0)\ }$}}&&{\makebox[0cm]{${\scriptstyle \left(\!\!\!\begin{array}{ccc}
\scriptstyle 0&\!\!\scriptstyle \Id_{E_0}&\!\! \scriptstyle 0\\
\scriptstyle D_F&\!\!\scriptstyle 0&\!\! \scriptstyle \Psi_1
\end{array}\!\!\!\right)\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle \left(\!\!\!\begin{array}{ccc}
\scriptstyle 0&\!\!\scriptstyle \Id_{E_0}&\!\! \scriptstyle 0\\
\scriptstyle D_F&\!\!\scriptstyle 0&\!\! \scriptstyle \Psi_1
\end{array}\!\!\!\right)\ }$}}&&{\makebox[0cm]{${\scriptstyle\Id_{E_1}\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\Id_{E_1}\ }$}}\\[12pt]
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&F_0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&E_0\oplus (\Psi_\bullet)_! C
&\stackrel{(D_E,\Phi_1)}{{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}}
&(\Phi_\bullet)_!(\Psi_\bullet)_! C
&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0\\[-3pt]
&&&\left(\!\!\!\begin{array}{c}
\scriptstyle\Phi_0\\ \scriptstyle -D_F
\end{array}\!\!\!\right)\\[-15pt]
&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}\\[12pt]
&&0&&0
\end{array}$$ We claim that rows and columns in this diagram are exact. The left vertical sequence is trivially exact. The upper howizontal sequence is exact by Proposition \[ex\_seq\_cones\] applied to $(\Phi_\bullet)_!$ and the cone $(\Psi_\bullet)_!(C)\subset F_1$. Exactness of the middle vertical sequence follows by adding a trivial $E_0$-term to the analogous sequence for $(\Psi_\bullet)_!$ and $C\subset G_1$. For the remaining middle horizontal sequence exactness of the enveloping sequence $$0\longrightarrow F_0\oplus G_0\longrightarrow
F_0\oplus E_0\oplus G_1 \longrightarrow E_1\longrightarrow 0$$ of linear spaces is easy to verify. Again by Proposition \[ex\_seq\_cones\] the preimage $\tilde C\subset F_0\oplus
E_0\oplus G_1$ of $(\Phi_\bullet\circ\Psi_\bullet)_!(C)\subset E_1$ intersects $0\oplus E_0\oplus G_1$ in $0\oplus E_0\oplus C$, and it is invariant under the action of $F_0$ on the first factor. Hence $\tilde
C= F_0\oplus E_0\oplus C$, proving exactness of the middle horizontal sequence.
Now exactness of the lower horizontal sequence and of the middle vertical sequence show that the preimage of $(\Phi_\bullet)_!(\Psi_\bullet)_! C$ under the composition of epimorphisms $F_0\oplus E_0\oplus G_1\to E_0\oplus F_1\to E_1$ from the lower right square equals $F_0\oplus E_0\oplus C$. This is the same as the preimage of $(\Phi_\bullet\circ \Psi_\bullet)_!C$. Therefore $(\Phi_\bullet)_!(\Psi_\bullet)_! C$ and $(\Phi_\bullet\circ\Psi_\bullet)_! C$ are the same cones in $E_1$.
The case of quasi-isomorphisms
------------------------------
By definition a morphism $\Phi_\bullet$ of two-term complexes is a quasi-isomorphism if $H^i(\Phi_\bullet)$ is an isomorphism for $i=0,1$. This is equivalent to requiring that $\Phi_\bullet$ viewed as a commutative square is cartesian and cocartesian, see Lemma \[ass\_exact\_sequence\]. Going up and down behaves well with respect to quasi-isomorphisms:
\[1-1-corr\] Let $\Phi_\bullet:F_\bullet\rightarrow E_\bullet$ be a quasi-isomorphism of two-term complexes of linear spaces with $F_0$ locally free. Then going up and down induces a functorial one-to-one correspondence between $F_0$-cones $C\subset F_1$ and $E_0$-cones $\bar C\subset E_1$.
[[*Proof.* ]{}]{}In view of Proposition \[left\_inverse\] it remains to show that if $\bar C\subset E_1$ is an $E_0$-cone then $\bar C=(\Phi_\bullet)_!
\Phi_\bullet^!(\bar C)$. This is a local problem. We may thus assume that there exists a local splitting $\sigma: E_0\oplus F_1\to F_0$ of the exact sequence $$0\longrightarrow F_0\longrightarrow E_0\oplus F_1
\stackrel{q}{\longrightarrow}E_1\longrightarrow0,\quad
q=D\circ \prj_1+\Phi_1\circ\prj_2$$ from Lemma \[ass\_exact\_sequence\]. Then $\chi=(\sigma,q):
E_0\oplus F_1\to F_0\oplus E_1$ is an isomorphism mapping the diagonal $F_0$-action on $E_0\oplus F_1$ to the action on the first factor of $F_0\oplus E_1$. Since $\sigma$ is a splitting, $\chi$ induces an isomorphism $\ker(\sigma)\to E_1$. Therefore $$\chi\big(\ker(\sigma)\cap q^{-1}(\bar C)\big)
=\chi(q^{-1}(\bar C))\cap (0\oplus E_1)=0\oplus \bar C.$$ But $\chi(q^{-1}(\bar C))$ is an $F_0$-cone, and hence Proposition \[quot\_cone\] implies $\chi(q^{-1}(\bar C)) = F_0\oplus
\bar C$. By definition this says $\bar C=(\Phi_\bullet)_!
\Phi_\bullet^!(\bar C)$.
Using the nice behavior under quasi-isomorphisms we may now define [*going down for morphisms in the derived category*]{} $D({{\mathop{\rm Coh}}}(X))$ of the category of coherent sheaves. In the language of linear spaces a morphism of two-term complexes in the derived category $\Phi_\bullet: F_\bullet\rightarrow E_\bullet$ consists of
1. another two-term complex $G_\bullet$
2. a quasi-isomorphism $\Theta_\bullet:E_\bullet\rightarrow G_\bullet$
3. and a morphism $\Psi_\bullet:F_\bullet\rightarrow G_\bullet$.
Two morphisms defined by tuples $(G_\bullet,\Theta_\bullet,
\Psi_\bullet)$ and $(G'_\bullet,\Theta'_\bullet,\Psi'_\bullet)$ are considered equivalent if there exists a two-term complex ${\tilde
G}_\bullet$ and quasi-isomorphisms $\Lambda_\bullet:
G_\bullet\rightarrow{\tilde G}_\bullet$, $\Lambda'_\bullet:
G'_\bullet\rightarrow{\tilde G}_\bullet$ making the following diagram commutative up to homotopy: $$\begin{array}{ccccc}
&&\ G_\bullet\ &&\\[7pt]
&{\makebox[0pt]{${\scriptstyle\Psi_\bullet\atop\ \ }{{\begin{picture}(0,0)
\unitlength 1pt\put(-15,-15){\vector(1,1){30}}\end{picture}}}\phantom{\scriptstyle\Psi_\bullet\atop\ \ }$}}&{\makebox[0cm]{$\phantom{\ \scriptstyle\raisebox{-5pt}{$\scriptstyle
\Lambda_\bullet$}}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle\raisebox{-5pt}{$\scriptstyle
\Lambda_\bullet$}}$}}&{\makebox[0pt]{$\phantom{\scriptstyle\Theta_\bullet\atop\ \ }
{{\begin{picture}(0,0)
\unitlength 1pt\put(15,-15){\vector(-1,1){30}}\end{picture}}}{\scriptstyle\Theta_\bullet\atop\ \ }$}}&\\[15pt]
\hspace{5cm}F_\bullet\ &&{\tilde G}_\bullet&&\ E_\bullet
\hspace{5cm}(*)\\[7pt]
&{\makebox[0pt]{${\ \ \atop\scriptstyle\Psi'_\bullet\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(-15,15){\vector(1,-1){30}}\end{picture}}}\phantom{\ \ \atop\scriptstyle\Psi'_\bullet\ }$}}&{\makebox[0cm]{$\phantom{\ \scriptstyle\raisebox{2pt}{$\scriptstyle
\Lambda'_\bullet$}}{{\begin{picture}(0,0)
\unitlength 1pt\put(0,-16){\vector(0,1){30}}\end{picture}}}{\ \scriptstyle\raisebox{2pt}{$\scriptstyle
\Lambda'_\bullet$}}$}}&{\makebox[0pt]{$\phantom{\ \ \atop\ \scriptstyle\Theta'_\bullet}
{{\begin{picture}(0,0)
\unitlength 1pt\put(15,15){\vector(-1,-1){30}}\end{picture}}}{\ \ \atop\ \scriptstyle\Theta'_\bullet}$}}&\\[15pt]
&&G'_\bullet&&
\end{array}$$
\[go\_down\_deriv\] (going down in the derived category) Let $\Phi_\bullet:F_\bullet
\rightarrow E_\bullet$ be a morphism of two-term complexes of linear spaces in the derived category, inducing an isomorphism on $H^0$ and a closed embedding on $H^1$. Moreover, we require $F_0$ to be locally free. When these assumptions are satisfied we say that [*going down is applicable to $\Phi_\bullet$*]{}. In this case the [*push-forward*]{} of an $F_0$-cone $C\subset F_1$ is defined to be the $E_0$-cone $$(\Phi_\bullet)_!(C):=(\Theta_\bullet)^!(\Psi_\bullet)_!(C)\subset
E_1\, ,$$ whenever $(G_\bullet,\Psi_\bullet,\Theta_\bullet)$ is a representative of $\Phi_\bullet$.
Using the previous results it is easy to check that this is well-defined.
\[coh\_cones\] One might wonder if there exists “going down” when being only given maps on the level of cohomology. There are three remarks I want to make on this.
1. A map in cohomology is considerably weaker than a map of complexes, even for two-term complexes. For instance, let $E_\bullet=[E_0\stackrel{D}{\rightarrow}E_1]$ be a non-split epimorphism of linear spaces and $K=\ker D$. Then $H^\bullet(E_\bullet) =H^\bullet([K_\bullet\rightarrow0])$, but the identity map in cohomology is not induced by a morphism of complexes.
2. There is going down for “cones coming from cohomology”: By such cones we mean cones of the form $C=p^{-1}(\bar C)$ for some $\bar C\subset H^1(E_\bullet)$, $p:E_1 \rightarrow H^1
(E_\bullet)$ the cokernel of $E_\bullet$. Namely, if $\rho: H^1(
E_\bullet)\rightarrow H^1(F_\bullet)$ is a closed embedding and $p': F_1 \rightarrow H^1(F_\bullet)$ is the cokernel of $F_\bullet$, one may set $\rho_!(C) :={p'}^{-1}(\rho(\bar C))$. In case $\rho=H^1(\Phi_\bullet)$ with $\Phi_\bullet:E_\bullet\rightarrow
F_\bullet$ a morphism to which going down is applicable, then $\rho_!(C)$ obviously coincides with $(\Phi_\bullet)_!(C)$.
3. Not every $E_0$-cone in $E_1$ comes from cohomology. As a simple example with $C\neq E_1$ but $H^1(E_\bullet)=0$ take $X=\Aff^1_k=\Spec k[T]$, $E_0=E_1=L({{\mathcal O}}_X) =\Spec k[T,U]$, $D:E_0\rightarrow E_1$ corresponding to the homomorphism of $k[T]$-algebras sending $U$ to $TU$, and $C=V(TU)\subset E_1$, the linear space corresponding to the structure sheaf of the origin.
See also Remark \[C\_X|M\_descend?\] for another, less artificial example.
Global normal cones
===================
If $\iota:X\hookrightarrow M$ is a closed embedding of algebraic $k$-schemes the normal cone $C_{X|M}\subset N_{X|M}$ is a $TM|_X$-cone (Example \[C\_Z|X\]). With nonsingular $M$ theses normal cones are essentially unique, namely up to vector bundle factors. In fact, if $\iota':X\hookrightarrow M'$ is another such embedding we may consider the diagonal $(\iota,\iota'):X
\hookrightarrow M\times M'$ to reduce to the case where $\iota=
\pi\circ\iota'$, $\pi:M'\rightarrow M$ a smooth morphism. But then there is an exact sequence of cones $$\begin{aligned}
0\longrightarrow(\iota')^*T_{M'|M}\longrightarrow C_{X|M'}
\longrightarrow C_{X|M}\longrightarrow0\, .\label{eqn1}\end{aligned}$$ Based on this observation Behrend and Fantechi show that to any $X$ there is associated a cone stack (a certain Artin stack) over $X$ of pure relative dimension zero, the [*intrinsic normal cone*]{} ${{\mathcal C}}_X$. Locally, ${{\mathcal C}}_X$ is nothing but the stack-theoretic quotient $C_{X|M}/T_M|_X$, and the above exact sequence of cones is responsible for the fact that these quotients glue.
One essential insight of Behrend and Fantechi is that one can retrieve an actual cone over $X$ by giving a morphism $\ph^\bullet:
{{\mathcal F}}^\bullet\rightarrow{{\mathcal L}}_X^\bullet$ in the derived category inducing an isomorphism in $H^0$ and an epimorphism in $H^{-1}$, and such that ${{\mathcal F}}^\bullet=[{{\mathcal F}}^{-1}\rightarrow{{\mathcal F}}^0]$ is a two-term complex of locally free sheaves. Here ${{\mathcal L}}_X^\bullet$ is the cotangent complex of $X$. (In the language of [@behrendfantechi], $\ph^\bullet$ is a “global resolution” of a “perfect obstruction theory” for $X$.)
The cotangent complex is a complicated and largely mysterious object canonically associated to any scheme, or even ringed topos [@illusie]. However, here we will work exclusively with the truncated complex $\tau_{\ge -1}{{\mathcal L}}_X^\bullet$. This is simply an object of the derived category that has the following explicit local description: If $U\subset X$ is an open subscheme and $U\hookrightarrow M$ is a closed embedding into a smooth scheme $M$ then $$\tau_{\ge -1} {{\mathcal L}}_X^\bullet= [{{\mathcal I}}/{{\mathcal I}}^2\to \Omega_M|_U],$$ where the complex on the right hand side has entries at $-1$ and $0$ (this follows from the exact triangle for the cotangent complex, see below). In particular, if $X$ is globally embedded into a smooth scheme we can avoid the cotangent complex at all.
Using our study of going up and down for $E$-cones we will see that the object needed is the following.
\[global\_normal\_space\] A [*global normal space*]{} for $X$ is a morphism $\ph^\bullet:
{{\mathcal F}}^\bullet = [{{\mathcal F}}^{-1}\rightarrow{{\mathcal F}}^0] \rightarrow
\tau_{\ge -1}{{\mathcal L}}_X^\bullet$ in the derived category with ${{\mathcal F}}^0$ locally free and inducing an isomorphism in $H^0$ and an epimorphism in $H^{-1}$.
Given a global normal space $\Phi_\bullet:\tau_{\le1}
(L_X)_\bullet\rightarrow F_\bullet$, now written in terms of linear spaces $\tau_{\le 1}(L_X)_\bullet=L(\tau_{\ge -1}{{\mathcal L}}_X^\bullet)$ etc., we may construct a cone $C=C(\Phi_\bullet)\subset F_0$ as follows: Let $U\subset X$ be an open set embedded into a nonsingular $M$, $\iota: U\hookrightarrow M$. The exact triangle of relative cotangent complexes associated to $U\rightarrow M\rightarrow\Spec k$ yields a morphism in the derived category $$\Lambda_\bullet:[T_M|_U\rightarrow N_{U|M}]\longrightarrow
\tau_{\le 1}(L_U)_\bullet$$ that induces isomorphisms in $H^i$, $i=0,1$. As $T_M|_U$ is a vector bundle the composition $\Phi_\bullet|_U\circ
\Lambda_\bullet$ fulfills the assumption of Definition \[go\_down\_deriv\]. We may thus define $$\begin{aligned}
C|_U\ :=\ \big(\Phi_\bullet|_U\circ\Lambda_\bullet\big)_!
(C_{U|M})\, .\label{eqn2}\end{aligned}$$ It remains to show
\[independence\] $(\Phi_\bullet|_U\circ\Lambda_\bullet)_!(C_{U|M})\subset F_1|_U$ is independent of choices.
[[*Proof.* ]{}]{}It suffices to treat the case of another embedding $\iota':U
\rightarrow M'$ s.th. $\iota=\pi\circ\iota'$ for some smooth morphism $\pi:M'\rightarrow M$, see above. We have a commutative diagram with exact rows and columns $$\begin{array}{cccccccccccl}
&&&&0&&0\\[8pt]
&&&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}\\[12pt]
&&&&\makebox[0pt]{${\iota'}^*T_{M'|M}$}&=\!=\!=&
\makebox[0pt]{${\iota'}^*T_{M'|M}$}\\[8pt]
&&&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}\\[12pt]
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&T_U&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&{\iota'}^*T_{M'}&
{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&N_{U|M'}
&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&T_1(U)&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0\\[8pt]
&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{$\phantom{\ \scriptstyle}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle}$}}&&{\makebox[0cm]{$\phantom{\ \scriptstyle}
{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}{\ \scriptstyle}$}}\\[12pt]
0&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&T_U&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&\iota^*T_M&
{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&N_{U|M}
&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&T_1(U)&{{\begin{picture}(25,6)(2.5,-4)
\unitlength 1pt\put(0,0){\vector(1,0){30}}\end{picture}}}&0&\!\!\!.\\[8pt]
&&&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}&&{\makebox[0cm]{${\scriptstyle\ }{{\begin{picture}(0,0)
\unitlength 1pt\put(0,16){\vector(0,-1){30}}\end{picture}}}\phantom{\scriptstyle\ }$}}\\[12pt]
&&&&0&&0
\end{array}$$ Here $T_1(U)$ is the linear space associated to the first higher cotangent sheaf of $U$. This shows that $D \pi$ induces a quasi-isomorphism $\Theta_\bullet: [T_{M'}|_U \rightarrow N_{U|M'}] \rightarrow
[T_M|_U\rightarrow N_{U|M}]$ with $\Lambda'_\bullet=
\Lambda_\bullet \circ\Theta_\bullet$. Moreover, from the exact sequence of cones (\[eqn1\]) $$C_{U|M'}\ =\ \Theta_1^{-1}(C_{U|M})\ =\ \Theta_\bullet^!(C_{U|M})\, .$$ Thus by Proposition \[1-1-corr\] we conclude $$(\Lambda'_\bullet)_!(C_{U|M'})\ =\
(\Lambda_\bullet)_!(\Theta_\bullet)_!\Theta_\bullet^!(C_{U|M})
\ =\ (\Lambda_\bullet)_!(C_{U|M})\, .$$
Here is the first of the two main results of this paper.
\[global\_normal\_cone\] Let $X$ be an algebraic $k$-scheme. To any global normal space $\Phi_\bullet: \tau_{\le 1}(L_X)_\bullet\rightarrow F_\bullet$ for $X$ is associated an $F_0$-cone $C(\Phi_\bullet)\subset F_1$, locally of the form (\[eqn2\]), and of pure dimension equal to $\rk F_0$.
[[*Proof.* ]{}]{}It remains to check the statement on the dimension. Locally, we may choose a representation of $\Phi_\bullet\circ \Lambda_\bullet:
[T_M|_U\rightarrow N_{U|M}] \rightarrow F_\bullet$, where $\iota:U\hookrightarrow M$ is a closed embedding of an open $U\subset X$, by $(G_\bullet,\Theta_\bullet,\Psi_\bullet)$ with
- $\Psi_\bullet:[T_M|_U\rightarrow N_{U|M}]\rightarrow G_\bullet$
- $\Theta_\bullet:F_\bullet\rightarrow G_\bullet$ a quasi-isomorphism
- $G_\bullet=[G_0\rightarrow G_1]$ with $G_0$ a vector bundle (!).
We get two exact sequences of cones (Proposition \[ex\_seq\_cones\]) $$\begin{array}{rcccccccl}
0&\longrightarrow&T_M|_U&\longrightarrow&G_0\oplus C_{U|M}
&\longrightarrow&(\Psi_\bullet)_!C_{U|M}&\longrightarrow&0\\[1ex]
0&\longrightarrow&F_0&\longrightarrow&G_0\oplus
C(\Phi_\bullet)|_U
&\longrightarrow&(\Psi_\bullet)_!C_{U|M}&\longrightarrow&0\, .
\end{array}$$ The first one shows that $(\Psi_\bullet)_!C_{U|M}$ is pure dimensional of dimension equal to $\rk G_0$, and then by the second one $C(\Phi_\bullet)$ is pure $(\rk F_0)$-dimensional.
$C(\Phi_\bullet)$ is called the [*global normal cone*]{} associated to the global normal space $\Phi_\bullet$.
\[C\_X|M\_descend?\] 1) The picture would be especially simple if for any closed embedding $\iota:X \hookrightarrow M$ into a nonsingular $M$, $C_{X|M}$ came from a cone in the intrinsically defined linear space $T_1(X)$. This is, however, generally wrong:
Consider the fat point $X=\Spec R$, $R=k[X,Y]/(X^2,XY,Y^2)$, with its embedding into $M=\Aff_k^2=\Spec k[X,Y]$. Letting $A$, $B$, $C$ correspond to the generators $X^2$, $XY$, $Y^2$ of the ideal, $C_{X|M}=\Spec R[A,B,C]/ (B^2-AC,XC-YB,XB-YA)$. $T_1(X)$ is the linear space corresponding to the kernel $I_T$ of $$\begin{aligned}
&R[A,B,C]/(XC-YB,XB-YA)\ \longrightarrow\ R[dX,dY],\\
&A\mapsto 2XdX,\ B\longmapsto YdX+XdY,\ C\mapsto 2YdY\, ,\end{aligned}$$ which is $(XA,YA,XB,YB,XC,YC)=(X,Y)\cdot(A,B,C)$. A cone in $N_{X|M}=\Spec R[A,B,C]/(XC-YB,XB-YA)$ comes from $T_1(X)$ iff its ideal is generated by polynomials in $XA$, $YA$, $XB$, $YB$, $XC$, $YC$. This is not the case for $B^2-AC$, and so $C_{X|M}$ does not come from a cone in $T_1(X)$.\
2) By uniqueness of minimal free resolutions of modules over a local ring (see e.g. [@eisenbud Thm.20.2]) it is not hard to show that for any, not necessarily closed point $x\in X$ there is a [*minimal*]{} germ of global normal spaces at $x$. This is constructed by embedding an étale neighborhood of $x$ in $X$ into a smooth $k$-scheme $M$ of dimension ${\rm embdim}_x X= \dim_{k(x)}
\Omega_x\otimes k(x)$. We assume $k$ perfect here to assure that regular $k$-schemes are smooth over $k$. A germ of global normal space at $x$ can then be defined by selecting a minimal set of generators for the ideal defining $X\hookrightarrow M$. This germ of global normal space is minimal in the sense that any other germ of global normal space at $x$ can be obtained by adding trivial factors.
As a consequence, the germ at $x$ of any global normal cone is isomorphic to $C_{X|M}$ plus a vector bundle factor. Morally speaking, the “nonlinear parts” of global normal cones are locally unique.
Virtual fundamental class and Fulton’s canonical class
======================================================
Virtual fundamental classes
---------------------------
If $X$ is an algebraic $k$-scheme and $\Phi_\bullet:\tau_{\le 1}
(L_X)_\bullet \rightarrow F_\bullet$ is a global normal space for $X$ with also [*$F_1$ a vector bundle*]{} we speak of a [*free*]{} global normal space of [*rank*]{} $\rk(\Phi_\bullet)=\rk F_0-\rk F_1$. We may then intersect the zero section $s:X\rightarrow F_1$ of $F_1$ with the global normal cone $C(\Phi_\bullet)\subset F_1$ to produce a class on $X$.
Let $\Phi_\bullet: \tau_{\le 1} (L_X)_\bullet\rightarrow F_\bullet$ be a free global normal space. The Chow class $$[X,\Phi_\bullet]\ :=\ s^![C(\Phi_\bullet)]\in A_{\rk(\Phi_\bullet)}(X)$$ is called [*virtual fundamental class*]{} of $X$ with respect to $\Phi_\bullet$.
Note that $[X,\Phi_\bullet]$ contains as much information as $[C(\Phi_\bullet)]\in A_{\rk(F_0)}(F_1)$, for $$[C(\Phi_\bullet)]\ =\ p^!s^![C(\Phi_\bullet)]\ =\ p^![X,\Phi_\bullet]\, ,$$ where $p:F_1\rightarrow X$ is the projection.
One of the most important property of such classes is their compatibility with specializations. In the application to the construction of invariants from moduli spaces associated to a projective manifold V, say (as in Gromov-Witten or Donaldson-theory), this property implies invariance under smooth deformations of $V$. There are two versions of the specialization theorem, one involving global normal spaces of the total space of a family, and the other working with [*relative*]{} global normal spaces (that is, a morphism $\Phi_\bullet: \tau_{\le 1}(L_{{{\mathcal X}}|S})_\bullet \rightarrow
F_\bullet$, where ${{\mathcal X}}\rightarrow S$ is the family of algebraic $k$-schemes under consideration). We do not have anything to add to the presentation in [@behrendfantechi Proposition 5.10 and Proposition 7.2], which translates almost literally into our language.
Before turning to an explicit formula for the computation of $[X,\Phi_\bullet]$ in hopefully more accessible terms, we want to add the following point of view: For any pure-dimensional cone $C$ in a vector bundle $F$ there is a formula for the intersection with the zero locus in terms of the Segre class of $C$ and the total Chern class of $F$ [@fulton Expl. 4.1.8]. Applied to $C(\Phi_\bullet)\subset
F_1$ it says $$[X,\Phi_\bullet]\ =\ \{c(F_1)\cap s(C(\Phi_\bullet))
\}_{\rk(\Phi_\bullet)}\, ,$$ where $\{\,.\,\}_d: A_*(X)\rightarrow A_d(X)$ denotes the projection to the $d$-dimensional part. Now for any $r>0$, the image of $[C(\Phi_\bullet)]$ under the monomorphism $\iota_r: F_1
\hookrightarrow F_1\oplus\Aff_X^r$ becomes rationally trivial, while $c(F_1\oplus \Aff^r_X) = c(F_1)$. Thus letting ${\tilde
s}^r:X\hookrightarrow F_1\oplus\Aff_X^r$ be the zero section we see $$\{c(F_1)\cap s(C(\Phi_\bullet))\}_{\rk(\Phi_\bullet)-r}\ =\
({\tilde s}^r)^!(\iota_r)_*[C(\Phi_\bullet)]\ =\ 0\, .$$ This teaches us two things: First, if $F_1$ splits off a trivial factor $F_1={\bar F}_1\oplus\Aff^1_X$ with $\img\Phi_1 \subset{\bar F}_1$ then $[X,\Phi_\bullet]=0$. So the result is trivial if $F_1$ is not chosen small enough. And second, if $[X,\Phi_\bullet]\neq 0$ then $\rk(\Phi_\bullet)$ is the smallest number $d$ such that $$\{c(F_1)\cap s(C(\Phi_\bullet))\}_d\ \neq\ 0\, .$$
Let $X$ be smooth of dimension $n$. Then the cotangent complex of $X$ is exact at ${{\mathcal L}}_X^{-1}$. So the natural morphism $\Phi_\bullet: \tau_{\le 1} (L_X)_\bullet\rightarrow
[T_X\rightarrow O]$, $O=X$ the trivial linear space over $X$, is an isomorphism in $H^0$ and $H^1$. Then $C(\Phi_\bullet)=X=O$ and $c(F_1)\cap s(C(\Phi_\bullet)) =s(C(\Phi_\bullet))=[X]$ has vanishing components in dimensions smaller than $n$.
Fulton’s canonical class
------------------------
If an algebraic $k$-scheme $X$ is globally embeddable into a smooth $k$-scheme $M$ (e.g. $X$ quasi-projective) then $$c_F(X)\ :=\ c(T_M|_X)\cap s(C_{X|M})\, \in A_*(X)$$ is a Chow-class on $X$ that is independent of the choice of embedding [@fulton Expl. 4.2.6].
\[fultons\_class\] $c_F(X)$ is called [*Fulton’s canonical class*]{}.
Note that if $X$ is smooth one may choose $X=M$ and so $c_F(X)=c(T_X)\cap[X]$. For comparison of $c_F(X)$ with Mather’s and MacPherson’s Chern classes see [@aluffi].
Given a (not necessarily free) global normal space $\Phi_\bullet:
\tau_{\le 1} (L_X)_\bullet\rightarrow F_\bullet$ for $X$, $c_F(X)$ can also be expressed as follows:
\[Fultons\_class\] Let $\Phi_\bullet:\tau_{\le 1} (L_X)_\bullet\rightarrow F_\bullet$ be a global normal space for a quasi-projective $X$. Then $$c_F(X)\ =\ c(F_0)\cap s(C(\Phi_\bullet))\, .$$
[[*Proof.* ]{}]{}By quasi-projectivity there exists a [global]{} closed embedding $\iota:X\hookrightarrow M$ of $X$ into a smooth $M$. This yields a globally defined morphism in the derived category $\Lambda_\bullet:
[T_M|_X\rightarrow N_{X|M}]\rightarrow \tau_{\le 1} (L_X)_\bullet$. Also by quasi-projectivity any sheaf is the quotient of a locally free sheaf. Hence there is a global representative $(G_\bullet,
\Theta_\bullet, \Psi_\bullet)$ of $\Phi_\bullet\circ \Lambda_\bullet$ in the construction of Theorem \[global\_normal\_cone\], that is, with $G_0$ a vector bundle, $\Theta_\bullet:F_\bullet \rightarrow
G_\bullet$ a quasi-isomorphism, $\Psi_\bullet: [T_M|_X\rightarrow
N_{X|M}]\rightarrow G_\bullet$. We get two exact sequences of cones with vector bundle kernels (see Proposition \[ex\_seq\_cones\]) $$\begin{array}{rcccccccl}
0&\longrightarrow&T_M|_X&\longrightarrow&
G_0\oplus C_{X|M}&\longrightarrow&
(\Psi_\bullet)_!(C_{X|M})&\longrightarrow&0\\[2ex]
0&\longrightarrow&F_0&\longrightarrow&G_0\oplus
C(\Phi_\bullet)&\longrightarrow&(\Psi_\bullet)_!(C_{X|M})
&\longrightarrow&0\, ,
\end{array}$$ which by the multiplicativity of Segre classes in exact sequences of cones with vector bundle kernels imply $$c(T_M|_X)\cap s(C_{X|M})\ =\ c(G_0)\cap s\Big((\Psi_\bullet)_!
(C_{X|M})\Big)\ =\ c(F_0)\cap s(C(\Phi_\bullet))\, .$$
If $X$ is any algebraic $k$-scheme with global normal spaces one could take the right-hand side of the formula in the proposition as definition for a generalization of Fulton’s canonical class on projective schemes. However, I was not able to prove independence of this class from the choice of $\Phi_\bullet$. And in case $X$ is not quasi-projective but embeddable into a smooth scheme, in the construction of Theorem \[global\_normal\_cone\] we might not be able to choose $G_0$ locally free. Then the coincidence of this class with $c_F(X)$ is not clear either. The problem is that on one hand the globally defined complex linking two global normal spaces $\Phi_\bullet: \tau_{\le 1}(L_X)_\bullet\rightarrow F_\bullet$, $\Phi'_\bullet: \tau_{\le 1} (L_X)_\bullet \rightarrow F'_\bullet$ is the cotangent complex, which need not be globally representable by a complex $L_\bullet$ with $L_0$ a vector bundle, while on the other hand Segre classes do not behave well in exact sequences unless the kernels are vector bundles.
We are now ready to deduce the announced formula for the virtual fundamental class.
\[closed\_formula\] Let $X$ be a projective $k$-scheme and $\Phi_\bullet: \tau_{\le 1}
(L_X)_\bullet\rightarrow F_\bullet$ a free global normal space for $X$ of constant rank $d$. Then $$[X,\Phi_\bullet]\ =\ \Big\{c(\ind F_\bullet)^{-1}\cap c_F(X)
\Big\}_d\, ,$$ where $\ind F_\bullet$ is the virtual bundle $F_0-F_1\in K^0(X)$.
[[*Proof.* ]{}]{}As remarked at the end of the last subsection the virtual fundamental class can be computed by the formula $$s^![C(\Phi_\bullet)]\ =\ \Big\{c(F_1)\cap
s(C(\Phi_\bullet))\Big\}_d\, .$$ Now just insert $c(F_0)^{-1}\cup c(F_0)$ and use Proposition \[Fultons\_class\]
1) This formula enlightens the dependence of virtual fundamental classes on the choice of global normal spaces: Interestingly, $[X,\Phi_\bullet]$ depends only on the index bundle of $F_\bullet$ rather than on any of the finer data used to construct $C(\Phi_\bullet)$. But note also that for another choice $\Phi'_\bullet: \tau_{\le 1} (L_X)_\bullet\rightarrow F'_\bullet$ of global normal space, $[X,\Phi'_\bullet]$ can not in general be computed from $[X,\Phi_\bullet]$ and $\ind F_\bullet$, $\ind
F'_\bullet$ alone.\
2) One can take this formula as [*definition*]{} of the virtual fundamental class of $X$ without knowing anything about the more sophisticated theory of global normal cones in the non-projective case. This was the point of view of the author in summer 1995 in an attempt to define GW-invariants in algebraic geometry, when I observed it from formal considerations. Unfortunately, I was not aware of Vistoli’s rational equivalence [@vistoli], from which the crucial independence of the invariants under smooth deformations can be derived. I learned also that the same formula has independently been discovered by Brussee for complex spaces constructed as zero locus of holomorphic Fredholm sections of holomorphic Banach bundles over complex Banach manifolds, as occurring for example in Seiberg-Witten theory [@brussee] (the interpretation of Brussee’s $c_*(X)$ as Fulton’s canonical class is not quite clear, though).\
3) At the beginning of Section 3 we mentioned Behrend and Fantechi’s intrinsic normal cone ${{\mathcal C}}_X$, which locally was the stack-theoretic quotient of $C_{X|M}$ by the action of $T_M|_X$ for some embedding $X\hookrightarrow M$ into a smooth space.
Now if $X$ is globally embedded into a smooth scheme $M$, ${{\mathcal C}}_X$ is globally the quotient of $C_{X|M}$ by $T_M|_X$. Hence in view of the multiplicative behavior of Segre classes in exact sequences of cones with vector bundle kernels, $c_F(X)$ could with some right considered as [*Segre class of ${{\mathcal C}}_X$*]{}.
Conversely, if there was a theory of Segre classes for cone stacks, the Segre class of ${{\mathcal C}}_X$ would generalize Fulton’s canonical class to arbitrary algebraic $k$-schemes.
[XXXXXX]{} P. Aluffi: [*Chern classes for singular hypersurfaces*]{}, Trans. Amer. Math. Soc. [**351**]{} (1999), 3989–4026 M. Artin: [*Versal deformations and algebraic stacks*]{}, Inv. Math. [**27**]{} (1974) 165–189 K. Behrend: [*GW-invariants in algebraic geometry*]{}, Inv. Math. [**127**]{} (1997), 601–617 K. Behrend, B. Fantechi: [*The intrinsic normal cone*]{}, Inv. Math. [**128**]{} (1997), 45–88 R. Brussee: [*The canonical class and the $C^\infty$-properties of Kähler surfaces*]{}, New York Journ. Math. [**2**]{} (1996), 103–146 A. Grothendieck: [*Éléments de géométrie algébrique II: Étude globale élémentaire de quelques classes de morphismes.*]{} Publ. Math. Inst. Hautes Étud. Sci. 8 (1961). D. Eisenbud: [*Commutative algebra. With a view toward algebraic geometry*]{}, Springer 1995 W. Fulton: [*Intersection theory*]{}, Springer 1984 L. Illusie: [*Complexe cotangent et déformations*]{}, 2 vols., Lecture Notes in Mathematics 239/283, Springer 1971/1972 R. Hartshorne: [*Residues and duality*]{}, Lecture Notes in Math.20, Springer 1966 J. Li, G. Tian: [*Virtual moduli cycles and GW-invariants of algebraic varieties*]{}, in: Topics in symplectic $4$-manifolds, 1st International Press lectures presented in Irvine, CA, USA, March 28-30, 1996 (R. Stern ed.), 47–83 B. Siebert: [*Gromov-Witten invariants for general symplectic manifolds*]{}, preprint [dg-ga 9608005]{}, accepted as Habilitationsschrift, Bochum 1998. B. Siebert: [*An update on (small) quantum cohomology*]{}, in: Mirror symmetry III, Proceedings of the conference on Geometry and Physics, Montreal 1995 (D.H. Phong, L. Vinet, S.T. Yau eds.), 279–312, AMS/IP Stud. Adv. Math. [**10**]{}, Amer. Math. Soc. 1999 B. Siebert: [*Algebraic and symplectic Gromov-Witten invariants coincide*]{}, Ann. Inst. Fourier [**49**]{} (1999) 1743–1795. B. Siebert: [*Logarithmic Gromov-Witten invariants*]{}, unfinished manuscript 2001. A. Vistoli: [*Intersection theory on algebraic stacks and on their moduli spaces*]{}, Inv. Math. [**97**]{} (1989) 613–670
[^1]: “Linear space (over $X$)” or “linear fiber space” (“Linearer Faserraum”) seem to be the classical notation for the “abelian cones” of [@behrendfantechi]
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Mestral
Mestral may refer to:
Armand Mestral (1917–2000), French actor and singer
George de Mestral (1907–1990), Swiss electrical engineer and inventor of the Velcro fabric fastener
Armand de Mestral, nephew of George, Swiss-Canadian academic, Order of Canada recipient
Mestral, a Vulcan character in the "Carbon Creek" episode of Star Trek: Enterprise, which mentioned Velcro | {
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Now I get to listen To aPPLE fanboys talk about how awesome the next iphone is going to be all damn day.
3,202 shares | {
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James Townsend (British politician)
James Townsend (baptised 8 February 1737 – 1 July 1787) was an English Whig politician and Lord Mayor of London in 1772–73. It has been claimed that he was the first person of mixed race to hold such positions.
Life and political career
James Townsend was baptised on 8 February 1737 at the church of St. Christopher-le-Stocks in London. He was the son of London merchant (and later MP) Chauncy Townsend and his wife Bridget Phipps. He attended Hertford College, Oxford in 1756.
In politics James Townsend was closely linked from the 1760s with the Whig grandee William Petty, 2nd Earl of Shelburne. Supported by Shelburne, he entered Parliament as Member for West Looe at a by-election in 1767, holding the seat until 1774.
In 1769, Townsend was elected alderman of the City of London for Bishopsgate ward and Sheriff of the City of London, becoming one of the leaders of the Whig party in London. In 1771 Townsend followed John Horne Tooke in breaking away from the Society of Gentlemen Supporters of the Bill of Rights, which had been created to support John Wilkes after his expulsion from the House of Commons, and of which Townsend was a co-founder. He turned from a friend of Wilkes's to one of his fiercest opponents. Townsend was elected Lord Mayor of London in 1772. Wilkes had come first in the polls but Sheriff Richard Oliver manipulated the voting process to prevent the election of Wilkes. This created political turmoil in the City and a mob incensed by Townsend's coup rioted outside Guildhall during the ball on Lord Mayor's Day. Townsend's arms were erased from the church of St Helen's Bishopsgate. John Wilkes was eventually elected Lord Mayor in 1774.
In 1781, Townsend presented a petition for electoral reform from the Tiverton activist Martin Dunsford. Townsend ran unsuccessfully for Parliament for the City of London in the general election of 1780, and in April 1782, Shelburne arranged for Townsend's election to Parliament for the pocket borough of Calne. As a member he backed some calls for reform, but mainly supported William Pitt the Younger. Shortly before his death in office he had opposed the impeachment of Warren Hastings.
Townsend died at his estate, Bruce Castle in Tottenham, on 1 July 1787. He was buried nearby at Old Church Tottenham in the mausoleum of his wife's family, the Coleraines. Her inheritance had made him a wealthy man.
Family
Townsend's mother Bridget (died 1762), who clandestinely married Chauncy Townsend in the Fleet Prison in 1730, was the daughter of James Phipps, who came from a prominent family of clothiers in Westbury, Wiltshire. At the age of sixteen, James Phipps entered the service of the Royal African Company of England (RAC) which traded slaves across the Atlantic between 1660 and 1752. Phipps lived on the Gold Coast for twenty years and died at Cape Coast Castle, the African headquarters of the RAC, in 1723. He became the highest-ranking RAC official in Africa before being removed from his post among accusations of embezzlement and abuse of power.
At Cape Coast James Phipps married Catherine, the daughter of an African woman and a white soldier in the service of the Dutch West India Company. In spite of being generously provided for in her husband's will, Catherine Phipps refused to move to England and died at Cape Coast in 1738. James and Catherine's children, including James Townsend's mother Bridget, were all of mixed race, so James Townsend has been claimed as Britain's first black member of parliament and as the first black Lord Mayor of London. It does not appear that this aspect of Townsend's family history was known at the time.
In 1763 James Townsend married Henrietta Rosa Peregrina du Plessis (1745–1785), the illegitimate daughter of Henry Hare, 3rd Baron Coleraine, and Rose du Plessis Henrietta Rosa was her father's heiress, but the estate escheated to the Crown because she was an alien. By means of his father's influence with Henry Fox, Townsend had the estate restored to him by private Act of Parliament. Bruce Castle, Townsend's house in Tottenham, was part of his wife's inheritance, and he redesigned parts of the building.
They had one son, Henry Hare, and one daughter, Henrietta Jemima. Henry Hare Townsend (1766–1827) married Charlotte Winter Lake, daughter of Sir James Lake, bart. and sister of Admiral Sir Willoughby Lake. Their son was the poet and writer Chauncy Hare Townshend (who spelt his surname thus) to whom Dickens dedicated Great Expectations. Henrietta Jemima Townsend (1764–1848) married Nicholas Owen Smythe Owen (1769–1804) of Condover Hall, Shropshire. They had no issue.
James Townsend's brother was the physician, scientist, and economist Joseph Townsend, who made important contributions to population studies and geology.
Notes
Category:1737 births
Category:1787 deaths
Category:Alumni of Hertford College, Oxford
Category:Members of the Parliament of Great Britain for English constituencies
Category:British MPs 1761–1768
Category:British MPs 1768–1774
Category:British MPs 1780–1784
Category:British MPs 1784–1790
Category:Sheriffs of the City of London
Category:18th-century lord mayors of London | {
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