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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Koszul Homology Localises

Statement

Let R be a commutative unital ring, let M be an R-module, let x be a finite sequence in R, and let S⊆R be a multiplicative subset. For every q, S−1Hq(K(x;M))≅Hq(K(x/1;S−1M)).

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Localises Termwise, Localisation of modules is exact.

Proof

technique · direct
1.1

Exact localization commutes with the kernel/image quotient defining homology.

givenalgebra
2.1

The termwise localization chain isomorphism identifies the result with the claimed localized Koszul homology.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

8 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