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Right derived functors form a cohomological delta functor

Statement

Assume the Axiom of Dependent Choice.

Let $A$ and $B$ be abelian categories, let $I$ be supplied injective resolution data on all objects of $A$ , and let $F : A \to B$ be an additive left exact functor. Then the additive functors $R_{I}^{n} F : A \to B, n \geq 0,$ admit connecting maps that make them into a cohomological delta functor on $A$ , and $R_{I}^{0} F$ is naturally isomorphic to $F$ .

Facts & Assumptions

Given: A short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ and an integer $n \geq 0$ .

[L1]

Each $R_{I}^{n} F$ is an additive functor (Right derived functors relative to supplied data are additive functors).

[L2]

The injective horseshoe is obtained by dualizing the projective horseshoe; the latter fits into a degreewise split short exact sequence of augmented complexes. Consequently the injective horseshoe also carries the dual degreewise split short exact sequence of cochain complexes (The horseshoe lemma for injective resolutions, The horseshoe lemma for projective resolutions).

[L3]

Applying $F$ to a short exact sequence of injective resolution complexes produces a long exact sequence in cohomology, natural under morphisms of such sequences (The long exact sequence in cohomology, Naturality of the cohomology connecting morphism).

[L4]

Replacing the supplied injective resolution datum at one object changes the derived functor only by natural isomorphism (Two supplied injective resolution data define naturally isomorphic right derived functors).

[L5]

The zeroth right derived functor of a left exact functor recovers the original functor (The zero-th right derived functor of a left exact functor recovers the functor).

[L6]

A cohomological delta functor is exactly the data listed in Cohomological delta functor.

[L7]

Different injective comparison extensions of the same object morphism induce the same maps on cohomology (The induced cohomology map is independent of the chosen injective comparison extension).

Proof

technique · direct

1.1L2L3L4givenconstruct

Choose the injective horseshoe from [L2]. Its degreewise split short exact sequence remains split exact after applying the additive functor $F$ , so it becomes a short exact sequence of cochain complexes in $B$ . Now [L3] gives its long exact cohomology sequence, and [L4] transports the middle cohomology groups to the fixed datum $I$ . This defines connecting maps $\partial^{n} : R_{I}^{n} F (A^{''}) \to R_{I}^{n + 1} F (A^{'}) .$

2.1L1L2L3L4L6L7step 1.1construct

Given a morphism of short exact sequences, use the degreewise biproduct form of the two injective horseshoes and construct compatible comparison maps by the dual comparison induction: injectivity extends the off-diagonal correction at each degree. By [L7], different comparison extensions induce the same maps on cohomology. Naturality of the cohomology connecting morphism in [L3] then makes the connecting squares commute, and [L1] supplies additivity. Therefore [L6] identifies $(R_{I}^{n} F, \partial^{n})$ as a cohomological delta functor on $A$ .

3.1L5step 2.1∎

The degree-zero term is naturally isomorphic to $F$ by [L5].

Right derived functors form a cohomological delta functor

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources