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Left derived functors relative to supplied data are additive functors

Statement

Assume the Axiom of Dependent Choice.

Let $P$ be a supplied projective resolution datum and $F : A \to B$ an additive functor between abelian categories. For every $n \in Z$ , the assignments $A ⟼ L_{n}^{P} F (A), u ⟼ L_{n}^{P} F (u)$ define an additive functor on the domain of $P$ .

Facts & Assumptions

Given: An integer $n$ .

[L1]

Left derived maps preserve identities (Left derived maps preserve identities).

[L2]

Left derived maps preserve composition (Left derived maps preserve composition).

[L3]

The category of complexes in an additive category is additive, so comparison lifts can be added degreewise (The category of complexes in an additive category is additive).

[L4]

Two comparison lifts of the same morphism are chain-homotopic (Projective comparison maps are unique up to chain homotopy).

[L5]

Applying $F$ degreewise preserves chain maps, and homology is an additive functor on complexes (An additive functor applies degreewise to complexes and chain maps, Homology is an additive functor).

[L6]

Chain-homotopic maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).

[L7]

An additive functor is a functor that is additive on each hom-group (Additive functor).

Proof

technique · direct

1.1

By [L1] and [L2], the assignments $A \mapsto L_{n}^{P} F (A)$ and $u \mapsto L_{n}^{P} F (u)$ already form a functor.

L1L2

1.2

Let $u, v : A \to B$ . Choose comparison lifts ${\tilde{u}}_{∙}$ and ${\tilde{v}}_{∙}$ . By [L3], their degreewise sum ${\tilde{u}}_{∙} + {\tilde{v}}_{∙}$ is again a chain map, and it lifts $u + v$ because augmentations are additive. Thus it is a comparison lift of $u + v$ .

L3givenconstruct

2.1

By definition of the derived map and [L5], $L_{n}^{P} F (u + v) = H_{n} ⁣ (F ({\tilde{u}}_{∙} + {\tilde{v}}_{∙})) = H_{n} ⁣ (F ({\tilde{u}}_{∙})) + H_{n} ⁣ (F ({\tilde{v}}_{∙})) .$ If a different comparison lift of $u + v$ were chosen, [L4] and [L6] would give the same homology map. Hence $L_{n}^{P} F (u + v) = L_{n}^{P} F (u) + L_{n}^{P} F (v) .$

L4L5L6step 1.2algebra

3.1

Step 1.1 gives functoriality, and step 2.1 gives additivity on each hom-group. Therefore [L7] identifies $L_{n}^{P} F$ as an additive functor.

L7step 1.1step 2.1∎

Depends on

Used by

Dependency tree · two levels

30 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)