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CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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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.

Regularity Notions Coincide Local Finite

Statement

Assume the Axiom of Choice. For finite M over Noetherian local (R,m) and x⊆m with nonzero terminal quotient, ordinary regularity, M-Koszul-regularity, and M-H1-regularity are equivalent.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and the rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Local Koszul Acyclicity Iff Regular Sequence, Koszul Regular Implies H One Regular, and H One Regular Local Implies Koszul Regular.

Proof

technique · direct
1.1givenalgebra

Regularity implies Koszul regularity, which implies H1-regularity.

2.1step 1.1algebra∎

Under the assumed AC, the local H1 implication and local acyclicity converse close the cycle. These applications are where Choice enters.

Depends on

Used by

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