False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: two syzygies of an object are canonically isomorphic

Statement

False. Two syzygies of an object are canonically isomorphic.

Facts & Assumptions

Given: The object $Z / 2 Z$ .

[L1]

Syzygies are relative to a displayed projective resolution (Syzygies and cosyzygies relative to a chosen resolution).

[L2]

Two projective resolutions give only stable isomorphism data for their syzygies (Syzygies from two projective resolutions are stably isomorphic).

Refutation

technique · direct

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 $0 \to Z \oplus Z \overset{(a, b) \mapsto (2 a, b)}{\to} Z \oplus Z \to Z / 2 Z \to 0,$ with augmentation $(x, y) \mapsto \overset{ˉ}{x}$ , has first syzygy $2 Z \oplus Z ≅ Z \oplus Z$ .

2.1L2step 1.1∎

The groups $Z$ and $Z \oplus Z$ are not isomorphic, so the two displayed syzygies are certainly not canonically isomorphic. This is exactly why [L2] stops at stable isomorphism rather than literal equality.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)