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.
The local depth-zero associated-prime criterion
Statement
Assume the Axiom of Choice. Let be a Noetherian local ring and let be a finite -module. Then
Facts & Assumptions
Given: The Axiom of Choice, a Noetherian local ring and a nonzero finite module.
Assuming the Axiom of Choice, Nakayama's lemma: Under the assumed Choice, if lies in the Jacobson radical and is finitely generated, then implies .
Depth zero and associated primes: Under the assumed Choice, if for a nonzero finite module over a Noetherian ring, then its depth with respect to is zero exactly when is contained in an associated prime.
Proof
Since , [F1] gives , so [F2] applies with . Its published proof uses finiteness of associated primes through an AC-qualified prime filtration; the assumed Choice supplies that premise. It says depth is zero exactly when for some associated prime .
Every associated prime is proper and every proper ideal of a local ring is contained in . Hence forces , proving both directions.
Depends on
Used by
- Depth is additive for a flat local homomorphism Corollary
- A parameter sequence that fails in a non-Cohen--Macaulay ring Example
- A normal Noetherian domain is the intersection of its height-one localizations Lemma
- depth two excludes finite punctured extension Lemma
- Nilpotent-orbit tensor covariants extend and polarize Lemma
- normal domain implies s two Lemma
- positive depth ring has regular minimal generator Lemma
- r one s two intersection of height one localisations Lemma
- Reflexive surface modules and codimension-one lattice extension Lemma
- serre r zero s one characterises reducedness Lemma
- Surface finite type normalization finite Lemma
- Finite étale covers extend across the closed point of a regular local ring Theorem
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
- Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)