TheoremStatement: Literature-sourcedProof: 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.

Left derived functors form a homological delta functor

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 $F : A \to B$ be an additive right exact functor. Then the additive functors $L_{n}^{P} F : A \to B, n \geq 0,$ together with the connecting maps of The connecting map for left derived functors, form a homological delta functor on $A$ . Moreover $L_{0}^{P} F$ is naturally isomorphic to $F$ .

Facts & Assumptions

Given: A short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ in $A$ and an integer $n > 0$ .

[L1]

Each $L_{n}^{P} F$ is an additive functor (Left derived functors relative to supplied data are additive functors).

[L2]

Item 9 supplies connecting maps from a chosen horseshoe construction, and item 10 makes them independent of that choice (The connecting map for left derived functors, The left derived connecting map is independent of the horseshoe resolution and lifts).

[L3]

The long exact homology sequence of a short exact sequence of complexes is natural (The long exact homology sequence is natural).

[L4]

The zeroth left derived functor of a right exact functor recovers the original functor (The zero-th left derived functor of a right exact functor recovers the functor).

[L5]

A homological delta functor is exactly the data listed in Homological delta functor.

Proof

technique · direct

1.1L2L3givenconstruct

By [L2], choose any horseshoe middle resolution for the given short exact sequence and define the connecting maps from its long exact homology sequence. Using [L3], that horseshoe sequence yields an exact long sequence $\dots \to L_{n}^{P} F (A^{'}) \to L_{n}^{P} F (A) \to L_{n}^{P} F (A^{''}) \overset{\partial_{n}}{\to} L_{n - 1}^{P} F (A^{'}) \to \dots,$ and item 10 ensures that this sequence depends only on the original short exact sequence, not on the auxiliary horseshoe data.

2.1L1L2L3L5step 1.1algebra

For a morphism of short exact sequences in $A$ , the generalized comparison assertion in [L2] supplies a compatible morphism between chosen horseshoe sequences. Apply [L3] to it. The connecting squares commute on the horseshoe level, and [L2] transports that naturality to the fixed datum $P$ . Together with the additivity from [L1], this is exactly the homological delta-functor structure required by [L5].

3.1L4step 2.1∎

The degree-zero term is naturally isomorphic to $F$ by [L4]. Hence the left derived functors form a homological delta functor with degree zero equal to the original right exact functor.

Left derived functors form a homological delta functor

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources