DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

serre r k and s k conditions

Definition

For a commutative Noetherian ring $R$ and an integer $j \geq 0$ , condition $(R_{j})$ means that $R_{p}$ is regular whenever $ht p \leq j$ . Condition $(S_{j})$ means that $depth R_{p} \geq \min {j, \dim R_{p}}$ for every prime $p$ . A finite module $M$ satisfies $(S_{j})$ if ${depth}_{R_{p}} M_{p} \geq \min {j, \dim {Supp}_{R_{p}} M_{p}}$ for every prime in its support. Outside the support the condition is vacuous, consistent with depth of the zero module being $+ \infty$ and the empty support having no nonnegative dimension. Thus the zero module satisfies all $(S_{j})$ conditions, and the zero ring satisfies both families vacuously.

Depends on

Used by

regular local ring satisfies r one Corollary
regular local ring satisfies s two Corollary
normal domain implies r one Lemma
normal domain implies s two Lemma
r one s two intersection of height one localisations Lemma
serre r zero s one characterises reducedness Lemma

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

Definition 10.157.1 (standard reference, not scraped)