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.

Depends on

Used by

Extending a degree-zero natural transformation Example
Two universal delta functors and their unique isomorphism Example
FALSE: universality removes the need for supplied resolution data False statement
An exact base functor has the trivial universal delta functor Proposition
Classical derived functors are the cohomology objects of the total derived functor Proposition
Satellites give the first derived functor Proposition
Universality is the construction-independence principle Remark

Dependency tree · two levels

34 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.12: Cohomological delta-functors (standard reference, not scraped)