feat: port CategoryTheory.Abelian.Ext (#4401)

Apologies about the git history, a whole chain of PRs have been opened before the previous one was merged.



Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
Co-authored-by: Johan Commelin <johan@commelin.net>
Co-authored-by: Kevin Buzzard <k.buzzard@imperial.ac.uk>
Co-authored-by: Joël Riou <joel.riou@universite-paris-saclay.fr>
Co-authored-by: Matthew Ballard <matt@mrb.email>

Mathlib.lean

@@ -583,6 +583,7 @@ import Mathlib.Analysis.Subadditive
 import Mathlib.CategoryTheory.Abelian.Basic
 import Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four
 import Mathlib.CategoryTheory.Abelian.Exact
+import Mathlib.CategoryTheory.Abelian.Ext
 import Mathlib.CategoryTheory.Abelian.FunctorCategory
 import Mathlib.CategoryTheory.Abelian.Generator
 import Mathlib.CategoryTheory.Abelian.Homology

Mathlib/CategoryTheory/Abelian/Ext.lean

@@ -0,0 +1,80 @@
+/-
+Copyright (c) 2021 Scott Morrison. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Morrison, Adam Topaz
+
+! This file was ported from Lean 3 source module category_theory.abelian.ext
+! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.Category.ModuleCat.Abelian
+import Mathlib.CategoryTheory.Functor.LeftDerived
+import Mathlib.CategoryTheory.Linear.Yoneda
+import Mathlib.CategoryTheory.Abelian.Opposite
+import Mathlib.CategoryTheory.Abelian.Projective
+
+/-!
+# Ext
+
+We define `Ext R C n : Cᵒᵖ ⥤ C ⥤ Module R` for any `R`-linear abelian category `C`
+by (left) deriving in the first argument of the bifunctor `(X, Y) ↦ Module.of R (unop X ⟶ Y)`.
+
+## Implementation
+
+It's not actually necessary here to assume `C` is abelian,
+but the hypotheses, involving both `C` and `Cᵒᵖ`, are quite lengthy,
+and in practice the abelian case is hopefully enough.
+
+PROJECT: State the alternative definition in terms of
+right deriving in the second argument, and show these agree.
+-/
+
+
+noncomputable section
+
+open CategoryTheory
+
+variable (R : Type _) [Ring R] (C : Type _) [Category C] [Abelian C] [Linear R C]
+  [EnoughProjectives C]
+
+/-- `Ext R C n` is defined by deriving in the first argument of `(X, Y) ↦ Module.of R (unop X ⟶ Y)`
+(which is the second argument of `linearYoneda`).
+-/
+
+-- Porting note: the lemmas `Ext_obj` and `Ext_map` generated by `@[simps]` were
+-- being rejected by the type-checking linter; it's unclear exactly why.
+-- In any case, these lemmas were not actually used downstream in mathlib3,
+-- and seem unlikely to be directly useful (rather than lemmas in terms of resolutions),
+-- and so have been removed during porting:
+-- @[simps! obj map]
+
+-- Porting note: the mathlib3 proofs of `map_id` and `map_comp` were timing out,
+-- but `aesop_cat` is fast if we leave them out.
+def Ext (n : ℕ) : Cᵒᵖ ⥤ C ⥤ ModuleCat R :=
+  Functor.flip
+    { obj := fun Y => (((linearYoneda R C).obj Y).rightOp.leftDerived n).leftOp
+      -- Porting note: if we use dot notation for any of
+      -- `NatTrans.leftOp` / `NatTrans.rightOp` / `NatTrans.leftDerived`
+      -- then `aesop_cat` can not discharge the `map_id` and `map_comp` goals.
+      -- This should be investigated further.
+      map := fun f =>
+        NatTrans.leftOp (NatTrans.leftDerived (NatTrans.rightOp ((linearYoneda R C).map f)) n) }
+set_option linter.uppercaseLean3 false in
+#align Ext Ext
+
+open ZeroObject
+
+/-- If `X : C` is projective and `n : ℕ`, then `Ext^(n + 1) X Y ≅ 0` for any `Y`. -/
+def extSuccOfProjective (X Y : C) [Projective X] (n : ℕ) :
+    ((Ext R C (n + 1)).obj (Opposite.op X)).obj Y ≅ 0 :=
+  let E := (((linearYoneda R C).obj Y).rightOp.leftDerivedObjProjectiveSucc n X).unop.symm
+  E ≪≫
+    { hom := 0
+      inv := 0
+      hom_inv_id := by
+        let Z : (ModuleCat R)ᵒᵖ := 0
+        rw [← (0 : 0 ⟶ Z.unop).unop_op, ← (0 : Z.unop ⟶ 0).unop_op, ← unop_id, ← unop_comp]
+        aesop }
+set_option linter.uppercaseLean3 false in
+#align Ext_succ_of_projective extSuccOfProjective