PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

An exact sequence is a complex, and its exactness agrees with the earlier notion

Statement

Every exact sequence in an abelian category is a chain complex. At each object, its exactness as a chain complex is exactly the previously defined exactness of the sequence.

Facts & Assumptions

Given: A composable sequence in an abelian category.

[L1]

An exact sequence is exact at each interior object in the sense of Exact sequence and short exact sequence in an abelian category.

[L2]

Exactness at a node is the equality of image and kernel subobjects (Exactness at a node).

[L3]

For composable morphisms $A \overset{f}{\to} B \overset{g}{\to} C$ , the relation $g f = 0$ is equivalent to the factorization of $im (f)$ through $\ker (g)$ (The arrow-theoretic criterion for exactness).

[L4]

Exactness of a chain complex at degree $n$ is the same equality of boundary and cycle subobjects at that degree (Exactness of a complex at a degree and acyclic complexes).

Proof

technique · direct

1.1

If $A \overset{f}{\to} B \overset{g}{\to} C$ is exact at $B$ , then [L1] and [L2] identify $im (f)$ with $\ker (g)$ . By [L3], this implies $g f = 0$ . Applying this at each consecutive pair shows that any exact sequence is a chain complex.

L1L2L3

2.1

At a chosen object of that exact sequence, the chain-complex notion in [L4] again compares $im (f)$ with $\ker (g)$ . That is exactly the criterion in [L2], so no new notion of exactness is introduced.

L2L4∎

Depends on

Used by

An exact short sequence as an acyclic three-term complex Example

Dependency tree · two levels

12 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, Remark 2.7.12 (standard reference, not scraped)