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

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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.

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Dimension shift for a homological delta functor effaced in the middle

Statement

Let $T = (T_{n}, \partial)$ be a homological delta functor. For a short exact sequence $0 \to K \to P \to A \to 0$ and an integer $n > 0$ , the connecting map $\partial_{n} : T_{n} (A) \to T_{n - 1} (K)$ has the following properties:

if $T_{n} (P) \to T_{n} (A)$ is the zero map, then $\partial_{n}$ is a monomorphism,
if $T_{n - 1} (K) \to T_{n - 1} (P)$ is the zero map, then $\partial_{n}$ is an epimorphism,
if both conditions hold, then $\partial_{n}$ is an isomorphism.

Facts & Assumptions

Given: A short exact sequence $0 \to K \to P \to A \to 0$ and an integer $n > 0$ .

[L1]

A homological delta functor attaches an exact segment $T_{n} (P) \to T_{n} (A) \overset{\partial_{n}}{\to} T_{n - 1} (K) \to T_{n - 1} (P)$ to the given short exact sequence (Homological delta functor).

Proof

technique · direct

1.1L1givenalgebra

By [L1], the kernel of $\partial_{n}$ is the image of $T_{n} (P) \to T_{n} (A)$ . Therefore if that incoming map is zero, then $\ker (\partial_{n}) = 0$ and $\partial_{n}$ is monic.

1.2L1givenalgebra

Again by [L1], the image of $\partial_{n}$ is the kernel of $T_{n - 1} (K) \to T_{n - 1} (P)$ . If the outgoing map is zero, then that kernel is all of $T_{n - 1} (K)$ , so $\partial_{n}$ is epic.

2.1step 1.1step 1.2∎

When both hypotheses hold, steps 1.1 and 1.2 show that $\partial_{n}$ is both monic and epic, hence an isomorphism in the abelian target category.

Depends on

Homological delta functor

Used by

Dependency tree · two levels

3 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

Romyar Sharifi, Homological Algebra (standard reference, not scraped)
Alexandre Grothendieck, Some aspects of homological algebra (Barr translation) (standard reference, not scraped)