CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ 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.

Two projective resolutions with nonisomorphic first syzygies

Statement refuted

Two projective resolutions of the same object must have isomorphic first syzygies.

Facts & Assumptions

Given: The object $Z / 2 Z$ .

[L1]

The standard cyclic-group resolution is projective (A projective resolution of a cyclic abelian group).

[L2]

First syzygies from two projective resolutions are only stably isomorphic in general (Syzygies from two projective resolutions are stably isomorphic).

Counterexample

1.1

The standard resolution $0 \to Z \overset{2}{\to} Z \to Z / 2 Z \to 0$ has first syzygy $2 Z ≅ Z$ . The stabilized resolution $0 \to Z \oplus Z \overset{(a, b) \mapsto (2 a, b)}{\to} Z \oplus Z \to Z / 2 Z \to 0$ is again projective and exact, and its first syzygy is $2 Z \oplus Z ≅ Z \oplus Z$ .

L1algebra

2.1

The groups $Z$ and $Z \oplus Z$ are not isomorphic, so the first syzygies of these two projective resolutions are not isomorphic. This is why [L2] stops at stable isomorphism.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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, An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 10.109: Rings of finite global dimension (standard reference, not scraped)