Right-(Left-)derived Functors

Fix an abelian category A, let J be a Δ-subcategory of K(A), let D_J be the corresponding derived category, and let

Q = Q_J : J → D_J

be the canonical Δ-functor. For any Δ-functors F and G from J to another Δ-category E, or from D_J to E, Hom(F, G) will denote the abelian group of Δ-functor morphisms from F to G.

A Δ-functor F : J → E is right-derivable if there exists a Δ-functor

RF : D_J → E

and a morphism of Δ-functors

ζ : F → RF ◦ Q

such that for every Δ-functor G : D_J → E the composed map

Hom(RF, G) →^natural Hom(RF ◦ Q, G ◦ Q) →^{via ζ} Hom(F, G ◦ Q)

is an isomorphism, (the map “via ζ” is an isomorphism). The Δ-functor F is left-derivable if there exists a Δ-functor

LF : D_J → E

and a morphism of Δ-functors

ζ : LF ◦ Q → F

such that for every Δ-functor G : D_J → E the composed map

Hom(G, LF) →^natural Hom(G ◦ Q, LF ◦ Q) →^{via ζ} Hom(G ◦ Q, F)

is an isomorphism (the map “via ζ” is an isomorphism).

Such a pair (RF, ζ) and (LF, ζ) are called the right-derived and left-derived functors of F respectively. Composition with Q gives an embedding of Δ-functor categories

Hom_∆(D_J, E)  ֒→ Hom_∆(J, E),

with image the full subcategory whose objects are the Δ-functors which transform quasi-isomorphisms into isomorphisms. Consequently we can regard a right-(left-)derived functor of F as an initial (terminal) object in the category of Δ-functor morphisms F → G′ (G′ → F ) where G′ ranges over all Δ-functors from J to E which transform quasi-isomorphisms into isomorphisms. As such, the pair (RF, ζ) (or (Lf, ζ)) – if it exists – is unique up to canonical isomorphism.

Let A′ be another abelian category. Any additive functor F : A → A′ extends to a Δ-functor F ̄ : K(A) → K(A′ ). Q′ : K(A′) → D(A′) being the canonical map, we will refer to derived functors of Q′F ̄, or of the restriction of Q′F ̄ to some specified Δ-subcategory J of K(A), as being “derived functors of F” and denote them by RF or LF.

Let A′ be an abelian category, and suppose that E is a Δ-subcategory of K(A′) or of D(A′). If RF exists we can set

RⁱF(A):= Hⁱ(RF(A)) (A ∈ J, i ∈ Z)

Since RF is a Δ-functor, any triangle A → B → C → A[1] in J is transformed by RF into a triangle in E, and hence we have an exact homology sequence:

…. → Rⁱ⁻¹F(C) → RⁱF(A) → RⁱF(B) → RⁱF(C) → Rⁱ⁺¹F(A) → ….

implying an exact sequence of A-complexes

0 → A → B → C → 0  (A, B, C ∈ J)

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