mathlib3 documentation


Ext #

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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.

theorem Ext_map (R : Type u_1) [ring R] (C : Type u_2) [category_theory.category C] [category_theory.abelian C] [category_theory.linear R C] [category_theory.enough_projectives C] (n : ) (c c' : Cᵒᵖ) (f : c c') :
(Ext R C n).map f = {app := λ (j : C), ((((category_theory.linear_yoneda R C).obj j).right_op.left_derived n).map f.unop).unop, naturality' := _}

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 linear_yoneda).


If X : C is projective and n : ℕ, then Ext^(n + 1) X Y ≅ 0 for any Y.