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.

Images and coimages of chain maps are computed degreewise

Statement

For a chain map $f : C_{∙} \to D_{∙}$ , the image complex and coimage complex in $Ch (A)$ are obtained degreewise from the images and coimages of the component maps $f_{n}$ .

Facts & Assumptions

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

[L1]

In any category with kernels and cokernels, the image is the kernel of the cokernel and the coimage is the cokernel of the kernel (Image and coimage in a category with kernels and cokernels).

[L2]

Kernels of chain maps are computed degreewise (The kernel of a chain map is computed degreewise).

[L3]

Cokernels of chain maps are computed degreewise (The cokernel of a chain map is computed degreewise).

Proof

technique · direct

1.1

By [L2] and [L3], the kernel and cokernel complexes of $f$ have $n$ th terms $\ker (f_{n})$ and $coker (f_{n})$ . Applying [L1] inside $Ch (A)$ therefore shows that the coimage and image complexes have $n$ th terms $coker (\ker f_{n})$ and $\ker (coker f_{n})$ .

L1L2L3

2.1

Those are exactly the ordinary coimage and image objects of $f_{n}$ by [L1]. Hence the complex-level image and coimage are computed degreewise, and the canonical coimage-to-image map is the family of the component canonical maps.

L1step 1.1∎

Depends on

Used by

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)