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.

source

@[simp]

theorem Ext_obj (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 : ℕ) (k : Cᵒᵖ) :

(Ext R C n).obj k = {obj := λ (j : C), opposite.unop ((((category_theory.linear_yoneda R C).obj j).right_op.left_derived n).obj (opposite.unop k)), map := λ (j j' : C) (f : j ⟶ j'), ((homotopy_category.homology_functor (Module R)ᵒᵖ (complex_shape.down ℕ) n).map ((homotopy_category.quotient (Module R)ᵒᵖ (complex_shape.down ℕ)).map ((category_theory.nat_trans.map_homological_complex (category_theory.nat_trans.right_op ((category_theory.linear_yoneda R C).map f)) (complex_shape.down ℕ)).app ((category_theory.projective_resolutions C).obj (opposite.unop k)).as))).unop, map_id' := _, map_comp' := _}

source

@[simp]

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 : 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' := _}

source

noncomputable def Ext (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 : ℕ) :

Cᵒᵖ ⥤ C ⥤ Module R

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).

Equations

Ext R C n = {obj := λ (Y : C), (((category_theory.linear_yoneda R C).obj Y).right_op.left_derived n).left_op, map := λ (Y Y' : C) (f : Y ⟶ Y'), category_theory.nat_trans.left_op (category_theory.nat_trans.left_derived (category_theory.nat_trans.right_op ((category_theory.linear_yoneda R C).map f)) n), map_id' := _, map_comp' := _}.flip

source

noncomputable def Ext_succ_of_projective (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] (X Y : C) [category_theory.projective X] (n : ℕ) :

((Ext R C (n + 1)).obj (opposite.op X)).obj Y ≅ 0

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

Equations

Ext_succ_of_projective R C X Y n = let E : opposite.unop ((((category_theory.linear_yoneda R C).obj Y).right_op.left_derived (n + 1)).obj X) ≅ opposite.unop 0 := (((category_theory.linear_yoneda R C).obj Y).right_op.left_derived_obj_projective_succ n X).unop.symm in E ≪≫ {hom := 0, inv := 0, hom_inv_id' := _, inv_hom_id' := _}