CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

The derived long exact sequence

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 $A$ , and let $I$ be supplied injective resolution data on all objects of $A$ .

If $F : A \to B$ is additive and right exact, then every short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ yields a natural long exact sequence $\dots \to L_{n}^{P} F (A^{'}) \to L_{n}^{P} F (A) \to L_{n}^{P} F (A^{''}) \to L_{n - 1}^{P} F (A^{'}) \to \dots \to L_{0}^{P} F (A^{''}) \to 0.$

If $G : A \to B$ is additive and left exact, then every such short exact sequence yields a natural long exact sequence $0 \to R_{I}^{0} G (A^{'}) \to R_{I}^{0} G (A) \to R_{I}^{0} G (A^{''}) \to R_{I}^{1} G (A^{'}) \to R_{I}^{1} G (A) \to \dots .$

Facts & Assumptions

Given: A short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ .

[L1]

Left derived functors form a homological delta functor (Left derived functors form a homological delta functor).

[L2]

Right derived functors form a cohomological delta functor (Right derived functors form a cohomological delta functor).

Proof

technique · direct

1.1L1given

The first displayed sequence is exactly the long exact sequence attached by [L1] to the given short exact sequence.

2.1L2step 1.1∎

The second displayed sequence is exactly the long exact sequence attached by [L2] to the same short exact sequence.

Depends on

Used by

Dependency tree · two levels

17 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)