LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.
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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

1.1L1L2givenconstruct

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 $d_{n}^{Q} q_{n} = q_{n - 1} d_{n}^{D} .$

2.1L2step 1.1algebradischarge-construct∎

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 $Ch (A)$ .

Depends on

Used by

Images and coimages of chain maps are computed degreewise Proposition
The quotient of the mapping cylinder by its source is the mapping cone 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)