PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01

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The cone connecting map agrees with the shifted identity up to the declared sign

Statement

For the canonical short exact sequence $0 \to D_{∙} \to Cone (f)_{∙} \to C [1]_{∙} \to 0$ of a chain map $f : C_{∙} \to D_{∙}$ in a module category, the connecting morphism $\partial_{n} : H_{n} (C [1]) \to H_{n - 1} (D)$ corresponds under the shift isomorphism $H_{n} (C [1]) \tilde{\to} H_{n - 1} (C)$ to the homology map $H_{n - 1} (f)$ . In particular, when $f = 1_{C}$ , the connecting morphism is the shifted identity up to the sign built into the shift convention.

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ of module complexes and an integer $n$ .

[L1]

The cone long exact sequence is obtained from the canonical short exact cone sequence (The cone long exact sequence).

[L2]

In module categories, the connecting map is computed by lifting a cycle and taking its boundary class (Elementwise formula for the connecting map in module categories).

[L3]

The shift isomorphism identifies $H_{n} (C [1])$ with $H_{n - 1} (C)$ using the sign convention fixed for shifts (Homology of a shift is shifted homology).

Proof

technique · direct

1.1L2L3givenalgebra

A class in $H_{n} (C [1])$ is represented by a cycle $x \in C_{n - 1}$ . In the canonical cone sequence, the element $(0, x) \in Cone (f)_{n}$ lifts that class, and its boundary is $(f_{n - 1} (x), 0) .$ By [L2], the connecting morphism sends $[x]$ to $[f_{n - 1} (x)] \in H_{n - 1} (D)$ .

2.1L1L3step 1.1algebra∎

Step 1.1 is exactly the formula for $H_{n - 1} (f)$ after identifying $H_{n} (C [1])$ with $H_{n - 1} (C)$ via [L3]. Hence the connecting map of the cone sequence agrees with $H_{n - 1} (f)$ under that shift identification. When $f = 1_{C}$ , this becomes the shifted identity with precisely the sign encoded in [L3].

Depends on

Used by

A nonnatural choice of connecting maps does not form a delta functor Counterexample
A degreewise split sequence with nonzero connecting map Example
FALSE: a degreewise split short exact sequence of complexes has zero connecting map False statement
FALSE: any sequence of functors with long exact sequences is a delta functor False statement
FALSE: the homology functor is exact on short exact sequences of complexes False statement

Dependency tree · two levels

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Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)