TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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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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A complex is exact at n exactly when its nth homology is zero

Statement

Let $C_{∙}$ be a chain complex in an abelian category and let $n \in Z$ . Then $C$ is exact at degree $n$ if and only if $H_{n} (C)$ is a zero object.

Facts & Assumptions

Given: A chain complex $C_{∙}$ and an integer $n$ .

[L1]

Exactness at degree $n$ means that the canonical map $β_{n} : B_{n} (C) \to Z_{n} (C)$ is an isomorphism (Exactness of a complex at a degree and acyclic complexes).

[L2]

The homology object $H_{n} (C)$ is the cokernel of $β_{n}$ (Homology object of a chain complex).

[L3]

In an abelian category, a morphism is epic exactly when its cokernel is zero (In an abelian category, monic means zero kernel and epic means zero cokernel).

[L4]

In an abelian category, a morphism that is both monic and epic is an isomorphism (An abelian category is balanced).

Proof

technique · direct

1.1

Assume $C$ is exact at degree $n$ . Then [L1] says $β_{n}$ is an isomorphism, hence in particular epic. By [L2] and [L3], the cokernel of $β_{n}$ , namely $H_{n} (C)$ , is therefore zero.

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2.1

Conversely, assume $H_{n} (C)$ is zero. By [L2], the cokernel of $β_{n}$ is zero, so [L3] makes $β_{n}$ epic. The map $β_{n}$ is monic because it factors the monic image inclusion of $B_{n} (C)$ through the monic cycle inclusion. Hence [L4] makes $β_{n}$ an isomorphism, and [L1] says that $C$ is exact at degree $n$ .

L1L2L3L4algebra∎

Depends on

Used by

FALSE: every infinite coproduct of complexes has homology equal to the coproduct of their homologies False statement

Dependency tree · two levels

15 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)
Romyar Sharifi, Homological Algebra, §2.7 (standard reference, not scraped)