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

The depth formula for flat local homomorphisms

Statement

Assume the Axiom of Choice. For every flat local homomorphism (R,m)→(S,n) of Noetherian local rings, depth⁡S=depth⁡R+depth⁡(S/mS). Moreover, S is Cohen--Macaulay if and only if R and the closed fibre are both Cohen--Macaulay. No finite-type hypothesis is required.

Facts & Assumptions

Given: The Axiom of Choice, the flat local Noetherian hypotheses and the closed fibre. Choice is spent through the regular-sequence lifting in Depth is additive for a flat local homomorphism and the dimension argument in The flat-local Cohen--Macaulay fibre criterion (The Axiom of Choice).

Proof

technique · direct
1.1given

The depth equality is Depth is additive for a flat local homomorphism under the stated Choice hypothesis.

2.1step 1.1algebra∎

The Cohen--Macaulay equivalence is The flat-local Cohen--Macaulay fibre criterion under the same Choice hypothesis. Both prerequisites use only the displayed ring hypotheses and AC, so no finite-type condition is introduced.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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