formal/lean/mathlib/ring_theory/flat.lean

/-
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