TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Derived functors are universal delta functors

Statement

Assume the Axiom of Dependent Choice.

Let $A$ and $B$ be abelian categories, let $P$ be supplied projective resolution data on all objects of $A$ , let $I$ be supplied injective resolution data on all objects of $A$ , and let $F : A \to B$ be additive.

If $F$ is right exact and the source category has enough projectives, then the left derived delta functor $(L_{n}^{P} F)$ is universal.

If $F$ is left exact and the source category has enough injectives, then the right derived delta functor $(R_{I}^{n} F)$ is universal.

Facts & Assumptions

Given: The stated exactness and enough-projectives or enough-injectives hypotheses.

[L1]

Left and right derived functors carry homological and cohomological delta functor structures (Left derived functors form a homological delta functor, Right derived functors form a cohomological delta functor).

[L2]

Their positive degrees are effaceable by projectives or injectives (Positive left derived functors are effaceable by projectives, Positive right derived functors are effaceable by injectives).

[L3]

Effaceable homological or cohomological delta functors are universal (Effaceable homological delta functors are universal, Effaceable cohomological delta functors are universal).

Proof

technique · direct

1.1L1L2L3given

In the right exact case, [L1] makes $(L_{n}^{P} F)$ a homological delta functor and [L2] makes it effaceable in positive degrees. Therefore [L3] shows that it is universal.

2.1L1L2L3step 1.1∎

In the left exact case, [L1] makes $(R_{I}^{n} F)$ a cohomological delta functor and [L2] makes it positively effaceable. Applying [L3] again gives its universality.

Derived functors are universal delta functors

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources