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 spectrum of a product ring is a disjoint union
Example
For commutative rings , there is a canonical isomorphism of locally ringed spaces
Facts & Assumptions
Given: Commutative rings and the idempotents , in .
A ring map induces a contraction map on prime spectra (The map of affine spectra induced by a ring homomorphism).
Verification
Since , a prime is uniquely either or for a prime of one factor.
The two families are disjoint open-and-closed sets, and projections give homeomorphisms with the two factor spectra by [F1].
Their localizations are and , so these homeomorphisms identify the structure sheaves.
Hence the spectrum is the claimed disjoint union of locally ringed spaces.
Depends on
- Affine schemes and their coordinate rings
- The map of affine spectra induced by a ring homomorphism
- The product ring $R \times S$ with componentwise operations, its identity $(1_R, 1_S)$ and its units $R^{\times} \times S^{\times}$
- Prime ideals and maximal ideals in a commutative ring
- The stalk of the affine structure sheaf at a prime is A_p
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
23 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
- The Stacks Project, Lemma 26.6.8 (standard reference, not scraped)