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 kernel of a chain map is computed degreewise

Statement

Let $f : C_{∙} \to D_{∙}$ be a chain map. The kernels $\ker (f_{n}) ↪ C_{n}$ assemble into a chain complex, and this complex is a kernel 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]

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

Proof

technique · constructive

1.1L1L2givenconstruct

Let $k_{n} : K_{n} \to C_{n}$ be a kernel of $f_{n}$ . By [L1], $f_{n - 1} d_{n}^{C} k_{n} = d_{n}^{D} f_{n} k_{n} = 0,$ so [L2] gives a unique differential $d_{n}^{K} : K_{n} \to K_{n - 1}$ with $k_{n - 1} d_{n}^{K} = d_{n}^{C} k_{n} .$

2.1L2step 1.1algebradischarge-construct∎

The family $K_{∙}$ is a chain complex because $k_{n - 2} d_{n - 1}^{K} d_{n}^{K} = d_{n - 1}^{C} d_{n}^{C} k_{n} = 0,$ and $k_{n - 2}$ is monic. The map $k : K \to C$ is a chain map by construction, and its componentwise kernel universal properties from [L2] together give the kernel universal property in $Ch (A)$ .

Depends on

Used by

A subcomplex is the kernel of its quotient map Proposition
Images and coimages of chain maps are computed degreewise Proposition
The category of complexes in an abelian category is abelian Theorem
The first isomorphism theorem for complexes 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)