LemmaStatement: 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 cokernel of a chain map is computed degreewise

Statement

Let $f : C_{∙} \to D_{∙}$ be a chain map. The cokernels $D_{n} ↠ coker (f_{n})$ assemble into a chain complex, and this complex is a cokernel of $f$ in $Ch (A)$ .

Facts & Assumptions

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

[L1]

A chain map satisfies $d_{n}^{D} f_{n} = f_{n - 1} d_{n}^{C}$ (Chain map).

[L2]

Cokernels are universal among arrows that kill the displayed map (Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers).

Proof

technique · constructive

1.1

Let $q_{n} : D_{n} \to Q_{n}$ be a cokernel of $f_{n}$ . Since $q_{n - 1} d_{n}^{D} f_{n} = q_{n - 1} f_{n - 1} d_{n}^{C} = 0$ by [L1], [L2] gives a unique differential $d_{n}^{Q} : Q_{n} \to Q_{n - 1}$ with $d_{n}^{Q} q_{n} = q_{n - 1} d_{n}^{D} .$

L1L2givenconstruct

2.1

The family $Q_{∙}$ is a chain complex because $d_{n - 1}^{Q} d_{n}^{Q} q_{n} = q_{n - 2} d_{n - 1}^{D} d_{n}^{D} = 0,$ and $q_{n}$ is epic. The quotient map $q : D \to Q$ is a chain map by construction, and the componentwise cokernel universal properties from [L2] assemble to the cokernel universal property in $Ch (A)$ .

L2step 1.1algebradischarge-construct∎

Depends on

Used by

Images and coimages of chain maps are computed degreewise Proposition
The category of complexes in an abelian category is abelian Theorem

Dependency tree · two levels

5 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)