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.
Cohen--Macaulay local modules and rings
Definition
Let be a Noetherian local ring and let be a nonzero finite -module. The module is Cohen--Macaulay when The zero module is excluded from this local definition. The local ring is Cohen--Macaulay when it is Cohen--Macaulay as an -module.
Depends on
Used by
- Cohen--Macaulayness and a regular parameter quotient Corollary
- Completion preserves Cohen--Macaulayness in both directions Corollary
- One regular system of parameters implies Cohen--Macaulayness Corollary
- The flat-local Cohen--Macaulay fibre criterion Corollary
- Zero-dimensional finite local modules are Cohen--Macaulay Corollary
- Maximal and global Cohen--Macaulay modules Definition
- Associated primes of a Cohen--Macaulay module have full dimension Lemma
- Localization preserves the Cohen--Macaulay depth--dimension equality Lemma
- The first parameter of a Cohen--Macaulay module is regular Lemma
Dependency tree · two levels
5 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)