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

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Dimension shift for a cohomological delta functor effaced in the middle

Statement

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

if $T^{n - 1} (I) \to T^{n - 1} (C)$ is the zero map, then $\partial^{n - 1}$ is a monomorphism,
if $T^{n} (A) \to T^{n} (I)$ is the zero map, then $\partial^{n - 1}$ is an epimorphism,
if both conditions hold, then $\partial^{n - 1}$ is an isomorphism.

Facts & Assumptions

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

[L1]

A cohomological delta functor attaches an exact segment $T^{n - 1} (I) \to T^{n - 1} (C) \overset{\partial^{n - 1}}{\to} T^{n} (A) \to T^{n} (I)$ to the given short exact sequence (Cohomological delta functor).

Proof

technique · direct

1.1L1givenalgebra

By [L1], the kernel of $\partial^{n - 1}$ is the image of $T^{n - 1} (I) \to T^{n - 1} (C)$ . If that map is zero, then $\ker (\partial^{n - 1}) = 0$ , so $\partial^{n - 1}$ is monic.

1.2L1givenalgebra

By [L1], the image of $\partial^{n - 1}$ is the kernel of $T^{n} (A) \to T^{n} (I)$ . If the latter map is zero, this kernel is all of $T^{n} (A)$ , so $\partial^{n - 1}$ is epic.

2.1step 1.1step 1.2∎

When both hypotheses hold, steps 1.1 and 1.2 show that $\partial^{n - 1}$ is an isomorphism.

Depends on

Cohomological 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

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