Skip to content

[Merged by Bors] - chore: semilinearize LinearMap.submodule[Co]map #39726

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Closed
ADedecker wants to merge 1 commit into from
Closed

[Merged by Bors] - chore: semilinearize LinearMap.submodule[Co]map #39726

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
23 changes: 12 additions & 11 deletions Mathlib/Algebra/Module/Submodule/Map.lean
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -629,17 +629,19 @@ end Submodule

namespace LinearMap

variable [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁]
variable [Semiring R] [Semiring R₂]
variable [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂]
variable {σ₁₂ : R →+* R₂} {σ₂₁ : R₂ →+* R}

/-- The `LinearMap` from the preimage of a submodule to itself.

This is the linear version of `AddMonoidHom.addSubmonoidComap`
and `AddMonoidHom.addSubgroupComap`. -/
@[simps!]
def submoduleComap (f : M →ₗ[R] M₁) (q : Submodule R M₁) : q.comap f →ₗ[R] q :=
def submoduleComap (f : M →ₛₗ[σ₁₂] M₂) (q : Submodule R₂ M₂) : q.comap f →ₛₗ[σ₁₂] q :=
f.restrict fun _ ↦ Submodule.mem_comap.1

theorem submoduleComap_surjective_of_surjective (f : M →ₗ[R] M₁) (q : Submodule R M₁)
theorem submoduleComap_surjective_of_surjective (f : M →ₛₗ[σ₁₂] M₂) (q : Submodule R₂ M₂)
(hf : Surjective f) : Surjective (f.submoduleComap q) := fun y ↦ by
obtain ⟨x, hx⟩ := hf y
use ⟨x, Submodule.mem_comap.mpr (hx ▸ y.2)⟩
Expand All @@ -653,23 +655,22 @@ This is the linear version of `AddMonoidHom.addSubmonoidMap` and `AddMonoidHom.a

TODO: Consider making this an `abbrev`, dropping its API, and renaming to something like
`restrictSubmodule`. -/
def submoduleMap (f : M →ₗ[R] M₁) (p : Submodule R M) : p →ₗ[R] p.map f :=
def submoduleMap [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) :
p →ₛₗ[σ₁₂] p.map f :=
f.restrict fun x hx ↦ Submodule.mem_map.mpr ⟨x, hx, rfl⟩

@[simp]
theorem submoduleMap_coe_apply (f : M →ₗ[R] M₁) {p : Submodule R M} (x : p) :
↑(f.submoduleMap p x) = f x := rfl
theorem submoduleMap_coe_apply [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) {p : Submodule R M}
(x : p) : ↑(f.submoduleMap p x) = f x := rfl

theorem submoduleMap_surjective (f : M →ₗ[R] M₁) (p : Submodule R M) :
theorem submoduleMap_surjective [RingHomSurjective σ₁₂] (f : M →ₛₗ[σ₁₂] M₂) (p : Submodule R M) :
Function.Surjective (f.submoduleMap p) := f.toAddMonoidHom.addSubmonoidMap_surjective _

@[grind inj]
theorem submoduleMap_injective {f : M →ₗ[R] M₁} (hf : Injective f) (p : Submodule R M) :
Injective (f.submoduleMap p) :=
theorem submoduleMap_injective [RingHomSurjective σ₁₂] {f : M →ₛₗ[σ₁₂] M₂} (hf : Injective f)
(p : Submodule R M) : Injective (f.submoduleMap p) :=
f.toAddMonoidHom.addSubmonoidMap_injective hf _

variable [Semiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {σ₂₁ : R₂ →+* R}

open Submodule

theorem map_codRestrict [RingHomSurjective σ₂₁] (p : Submodule R M) (f : M₂ →ₛₗ[σ₂₁] M) (h p') :
Expand Down
Loading