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/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import ring_theory.noetherian
/-!
# Flat modules
A module `M` over a commutative ring `R` is *flat*
if for all finitely generated ideals `I` of `R`,
the canonical map `I ⊗ M →ₗ M` is injective.
This is equivalent to the claim that for all injective `R`-linear maps `f : M₁ → M₂`
the induced map `M₁ ⊗ M → M₂ ⊗ M` is injective.
See <https://stacks.math.columbia.edu/tag/00HD>.
This result is not yet formalised.
## Main declaration
* `module.flat`: the predicate asserting that an `R`-module `M` is flat.
## TODO
* Show that tensoring with a flat module preserves injective morphisms.
Show that this is equivalent to be flat.
See <https://stacks.math.columbia.edu/tag/00HD>.
To do this, it is probably a good idea to think about a suitable
categorical induction principle that should be applied to the category of `R`-modules,
and that will take care of the administrative side of the proof.
* Define flat `R`-algebras
* Define flat ring homomorphisms
- Show that the identity is flat
- Show that composition of flat morphisms is flat
* Show that flatness is stable under base change (aka extension of scalars)
For base change, it will be very useful to have a "characteristic predicate"
instead of relying on the construction `A ⊗ B`.
Indeed, such a predicate should allow us to treat both
`polynomial A` and `A ⊗ polynomial R` as the base change of `polynomial R` to `A`.
(Similar examples exist with `fin n → R`, `R × R`, `ℤ[i] ⊗ ℝ`, etc...)
* Generalize flatness to noncommutative rings.
-/
universes u v
namespace module
open function (injective)
open linear_map (lsmul)
open_locale tensor_product
/-- An `R`-module `M` is flat if for all finitely generated ideals `I` of `R`,
the canonical map `I ⊗ M →ₗ M` is injective. -/
class flat (R : Type u) (M : Type v) [comm_ring R] [add_comm_group M] [module R M] : Prop :=
(out : ∀ ⦃I : ideal R⦄ (hI : I.fg), injective (tensor_product.lift ((lsmul R M).comp I.subtype)))
namespace flat
open tensor_product linear_map _root_.submodule
instance self (R : Type u) [comm_ring R] : flat R R :=
⟨begin
intros I hI,
rw ← equiv.injective_comp (tensor_product.rid R I).symm.to_equiv,
convert subtype.coe_injective using 1,
ext x,
simp only [function.comp_app, linear_equiv.coe_to_equiv, rid_symm_apply, comp_apply,
mul_one, lift.tmul, subtype_apply, algebra.id.smul_eq_mul, lsmul_apply]
end⟩
end flat
end module
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