Alphabeta MathHow 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.
Polynomial extensions of Cohen--Macaulay rings
Statement
Assume the Axiom of Choice. If is a Noetherian Cohen--Macaulay ring, then is Cohen--Macaulay for every .
Facts & Assumptions
Given: The Axiom of Choice and a Noetherian Cohen--Macaulay ring . Cohen--Macaulayness for a nonlocal ring is tested at all prime localizations.
Under AC, polynomial extension of a finite globally Cohen--Macaulay module preserves global Cohen--Macaulayness (Polynomial extension preserves Cohen--Macaulayness).
A polynomial ring in one variable over a Noetherian ring is Noetherian (Hilbert basis theorem: if is Noetherian then is Noetherian).
Proof
Under the stated AC, apply [L1] to the finite module ; this gives the one-variable polynomial ring as Cohen--Macaulay.
The one-variable output is again Noetherian by [L2], so iterating [L1] one variable at a time proves the assertion for every finite ; is the original ring.
Depends on
Used by
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
- Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)