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.
Support of a module
Definition
Let be a commutative ring and let be a left -module. The support of is
where ranges over the prime ideals of and denotes localisation at the multiplicative set (Localisation at a prime ideal: ).
On this page, before the separate spectrum page is built, is read simply as a set of prime ideals.
Depends on
Used by
- A finite module that vanishes at a prime vanishes on some principal neighbourhood of that prime Corollary
- The support of Z/12Z is the pair of primes (2) and (3) Example
- A prime lies in the support exactly when some element has annihilator inside it Lemma
- The support of a cyclic quotient is its vanishing set Lemma
- Support in a short exact sequence is the union of the outer supports Theorem
- Support of an arbitrary direct sum is the union of the supports Theorem
- Support under localisation is restriction to primes disjoint from the denominator set Theorem
Dependency tree · two levels
6 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
- A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., (13.26) (standard reference, not scraped)
- The Stacks Project, Section 10.40: Support (standard reference, not scraped)