TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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.

L1L2L3given

2.1

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.

L1L2L3step 1.1∎

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
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)