TheoremStatement: 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.

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The first isomorphism theorem for complexes

Statement

For every chain map $f : C_{∙} \to D_{∙}$ in an abelian category, there is a canonical isomorphism of complexes $C_{∙} / \ker (f) ≅ im (f) .$

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ .

[L1]

In an abelian category, $C_{n} / \ker (f_{n}) ≅ im (f_{n})$ canonically for every component map $f_{n}$ (First isomorphism theorem in an abelian category).

[L2]

The image and coimage of a chain map are computed degreewise (Images and coimages of chain maps are computed degreewise).

[L3]

The kernel of $f$ is the degreewise kernel complex (The kernel of a chain map is computed degreewise).

Proof

technique · direct

1.1

By [L3], the quotient complex $C / \ker (f)$ has $n$ th term $C_{n} / \ker (f_{n})$ . By [L1], each of these terms is canonically isomorphic to $im (f_{n})$ .

L1L3

2.1

By [L2], the complex $im (f)$ has $n$ th term $im (f_{n})$ , and the component isomorphisms from step 1.1 are the coimage-to-image comparisons of the chain map $f$ . Hence they commute with the differentials and assemble to the claimed isomorphism of complexes.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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)