DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Syzygies and cosyzygies relative to a chosen resolution

Definition

Fix a projective resolution $\dots \to P_{2} \overset{d_{2}}{\to} P_{1} \overset{d_{1}}{\to} P_{0} \overset{ε}{\to} A \to 0.$ Its first syzygy relative to this resolution is $Ω_{P}^{1} (A) : = \ker (ε),$ and for $n \geq 2$ its $n$ th syzygy relative to this resolution is $Ω_{P}^{n} (A) : = \ker (d_{n - 1}) .$

Dually, for an injective resolution $0 \to A \overset{η}{\to} I^{0} \overset{d^{0}}{\to} I^{1} \overset{d^{1}}{\to} I^{2} \to \dots,$ its first cosyzygy relative to this resolution is $Σ_{I}^{1} (A) : = coker (η),$ and for $n \geq 2$ its $n$ th cosyzygy relative to this resolution is $Σ_{I}^{n} (A) : = coker (d^{n - 2}) .$

These objects are attached to the displayed resolution. No canonical-object claim is made without further comparison data.

Syzygies and cosyzygies relative to a chosen resolution

Definition

Depends on

Dependency tree · 2 levels

Sources