CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29‡ rests on unproved material (inherited)

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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‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The kernel row of a morphism of short exact sequences need not be short exact

Statement refuted

For every morphism of short exact sequences in an abelian category, the induced kernel row is itself short exact.

Diagram

diagram failed to render:
0 \arrow[r] & \mathbb Z \arrow[r, "\times 2"] \arrow[d, "\times 2"'] & \mathbb Z \arrow[r] \arrow[d, "\times 2"'] & \mathbb Z/2 \arrow[r] \arrow[d, "0"'] & 0 \\
0 \arrow[r] & \mathbb Z \arrow[r, "\times 2"'] & \mathbb Z \arrow[r] & \mathbb Z/2 \arrow[r] & 0.

Facts & Assumptions

Given: In $A b$ , the commutative diagram above.

[L1]

The category $A b$ is abelian (Abelian groups form an abelian category).

[L2]

For such a diagram, the kernel row is exact at its first two nodes (The kernel row and cokernel row of a morphism of short exact sequences are exact at two nodes each).

Counterexample

technique · direct

1.1

Each row is short exact, and the vertical maps make a morphism of short exact sequences in the abelian category $A b$ by [L1].

L1givenalgebra

1.2

The kernels of the three vertical maps are $\ker (\times 2) = 0, \ker (\times 2) = 0, \ker (0) = Z / 2.$ So the kernel row is $0 \to 0 \to 0 \to Z / 2.$ By [L2], it is exact at the first two nodes.

L2step 1.1algebra

2.1

The last map in that row is the zero map $0 \to Z / 2$ , hence not epic. Therefore the kernel row is not short exact.

step 1.2algebra

3.1

This refutes the statement.

step 2.1∎

Depends on

Used by

The kernel row failure for multiplication by two computed Example
FALSE: the kernel row of a morphism of short exact sequences is short exact False statement

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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)