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

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.

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

1.1L1algebra

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 $2 Z \oplus Z ≅ Z \oplus Z$ .

2.1L2step 1.1∎

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.

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)