TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Naturality of the cohomology connecting morphism

Statement

A morphism of short exact sequences of cochain complexes induces a commutative square $\begin{array}{ccc} H^{n} (C) & \overset{\partial^{n}}{\to} & H^{n + 1} (A) \\ ↓ & ↓ \\ H^{n} (C^{'}) & \overset{\partial^{' n}}{\to} & H^{n + 1} (A^{'}) \end{array}$ for every $n \in Z$ .

Facts & Assumptions

Given: A morphism of short exact sequences of cochain complexes.

[L1]

A cochain complex is read as a chain complex by the grading-reversal convention $(C^{♯})_{n} = C^{- n}$ (Cochain complex in an abelian category).

[L2]

The homology connecting morphism is natural under morphisms of short exact sequences (Naturality of the homology connecting morphism).

[L3]

The cohomology connecting maps are the reindexed homology connecting maps (The long exact sequence in cohomology).

Proof

technique · direct

1.1L1givenconstruct

Apply [L1] to both rows of the given morphism. This turns the cochain ladder into a morphism of short exact sequences of chain complexes.

2.1L2L3step 1.1algebra∎

The resulting lower-index connecting square commutes by [L2]. Translating back with [L3] gives the upper-index square in the statement.

Depends on

Used by

Natural transformations of base functors give morphisms of derived delta functors Proposition
The two Ext long exact sequences agree under balance Proposition
Naturality of singular mayer vietoris connectors Theorem
Right derived functors form a cohomological delta functor Theorem
The long exact Ext sequence in the first variable Theorem

Dependency tree · two levels

11 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)