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The acyclic-resolution theorem for left derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum on a class $D$ , let $F : A \to B$ be an additive right exact functor, and let $\dots \to Q_{2} \to Q_{1} \to Q_{0} \to A \to 0$ be an $F$ -acyclic resolution of $A$ relative to $P$ . Assume moreover that $A \in D$ and that, for $Z_{0} : = A$ and $0 \to Z_{q + 1} \to Q_{q} \to Z_{q} \to 0 (q \geq 0),$ each $Z_{q}$ lies in $D$ . Then for every $n \geq 0$ there is a canonical isomorphism $L_{n}^{P} F (A) \tilde{\to} H_{n} (F (Q_{∙}_{d e l})) .$

Facts & Assumptions

Given: An $F$ -acyclic resolution $\dots \to Q_{1} \to Q_{0} \to A \to 0$ of $A$ relative to $P$ , the associated objects $Z_{q} \in D$ , and an integer $n \geq 0$ .

[L1]

An $F$ -acyclic resolution is an exact augmented complex whose terms are $F$ -acyclic objects (An F-acyclic resolution, An acyclic object for a right exact functor).

[L2]

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

[L3]

Change of supplied projective resolution data produces canonical natural isomorphisms of left derived functors (Two supplied projective resolution data define naturally isomorphic left derived functors).

[L4]

Projective resolutions of a short exact sequence can be arranged into a short exact sequence of chain complexes by the projective horseshoe lemma (The horseshoe lemma for projective resolutions).

[L5]

A short exact sequence of chain complexes yields a long exact sequence in homology (The long exact sequence in homology).

Proof

technique · direct

1.1L1givenconstruct

By exactness in [L1], let $Z_{0} = A$ and for each $q \geq 0$ let $Z_{q + 1}$ fit into a short exact sequence $0 \to Z_{q + 1} \to Q_{q} \to Z_{q} \to 0.$ Every $Q_{q}$ is $F$ -acyclic by [L1].

2.1L2L3L4L5step 1.1algebra

Apply [L4] to each short exact sequence from step 1.1 using the supplied projective resolutions of $Z_{q + 1}$ and $Z_{q}$ , which exist by the domain hypothesis in the statement. The middle projective resolution from horseshoe need not be the supplied one for $Q_{q}$ , but [L3] identifies the resulting left derived objects. After applying $F$ and [L5], the higher homology of the middle term vanishes because $Q_{q}$ is $F$ -acyclic, while [L2] identifies the degree-zero term. Thus we obtain exact sequences $0 \to L_{1}^{P} F (Z_{q}) \to F (Z_{q + 1}) \to F (Q_{q}) \to F (Z_{q}) \to 0$ and isomorphisms $L_{m}^{P} F (Z_{q + 1}) \tilde{\to} L_{m + 1}^{P} F (Z_{q}) (m > 0) .$

3.1step 2.1algebra

Repeatedly applying the isomorphisms from step 2.1 gives $L_{n}^{P} F (A) = L_{n}^{P} F (Z_{0}) \tilde{\to} L_{1}^{P} F (Z_{n - 1}) (n > 0) .$

4.1step 2.1step 3.1algebra

For $n > 0$ , the exact sequence $F (Z_{n + 1}) \to F (Q_{n}) \to F (Z_{n}) \to 0$ from step 2.1 shows that the quotient of $F (Q_{n})$ by boundaries is $F (Z_{n})$ , and the same step identifies the kernel of $F (Z_{n}) \to F (Q_{n - 1})$ with $L_{1}^{P} F (Z_{n - 1})$ . Therefore $H_{n} (F (Q_{∙}_{d e l})) ≅ L_{1}^{P} F (Z_{n - 1}) ≅ L_{n}^{P} F (A) .$

5.1L2step 2.1step 4.1∎

For $n = 0$ , right exactness gives $F (Q_{1}) \to F (Q_{0}) \to F (A) \to 0$ , so $H_{0} (F (Q_{∙}_{d e l})) ≅ F (A)$ . By [L2], $F (A) ≅ L_{0}^{P} F (A)$ . Together with step 4.1, this proves the theorem for all $n \geq 0$ .

Depends on

Used by

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)