ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

A two-term complex has kernel and cokernel homology

Example

Let $R$ be a ring and let $f : M \to N$ be a homomorphism. Regard $0 \to M \overset{f}{\to} N \to 0$ as a chain complex with $M$ in degree $1$ and $N$ in degree $0$ . Then $H_{1} ≅ \ker (f), H_{0} ≅ coker (f),$ and all other homology objects are zero.

Facts & Assumptions

Given: A module homomorphism $f : M \to N$ .

[L1]

Module categories are abelian (Modules over a ring form an abelian category).

[L2]

Cycles, boundaries, and homology are defined by kernels, images, and the quotient $Z_{n} / B_{n}$ (Cycle and boundary subobjects of a complex, Homology object of a chain complex).

Verification

technique · direct

1.1L1L2givenalgebra

In degree $1$ , the outgoing differential is $f$ , so $Z_{1} = \ker (f)$ and $B_{1} = 0$ . In degree $0$ , the incoming differential is $f$ and the outgoing one is $0$ , so $Z_{0} = N$ and $B_{0} = im (f)$ . Every other term is zero.

2.1L2step 1.1∎

Therefore [L2] gives $H_{1} = \ker (f), H_{0} = N / im (f) = coker (f),$ and $H_{n} = 0$ for $n \neq 0, 1$ .

Depends on

Used by

The multiplication-by-m complex computes a cyclic group Example

Dependency tree · two levels

12 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

Romyar Sharifi, Homological Algebra, §2.7 (standard reference, not scraped)