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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ 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.

Hom is not exact

Statement refuted

For every object $X$ in an abelian category, the functor $A (-, X)$ is exact.

Facts & Assumptions

Given: In $A b$ , the short exact sequence $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 \to 0.$

[L1]

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

[L2]

Hom is left exact, but no right exactness was asserted (Hom is left exact in each variable).

Counterexample

technique · direct

1.1

Apply $A b (-, Z)$ to the given short exact sequence. This yields $0 \to Hom (Z / 2, Z) \to Hom (Z, Z) \overset{(\times 2)^{*}}{\to} Hom (Z, Z)$ , which is left exact by [L2].

L1L2given

2.1

The group $Hom (Z, Z)$ is $Z$ , and precomposition with $\times 2$ is multiplication by $2$ on that copy of $Z$ . This map is not surjective, so the Hom sequence is not exact at the right-hand term.

step 1.1algebra

3.1

Therefore the contravariant Hom functor need not be exact.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)