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.

Depends on

Used by

Syzygies from two projective resolutions are stably isomorphic Corollary
FALSE: two syzygies of an object are canonically isomorphic False statement
The horseshoe kernel fits into a short exact sequence Lemma

Dependency tree · two levels

9 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)