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.
Morphisms of locally ringed spaces
Definition
Let be a morphism of ringed spaces between locally ringed spaces. It is a morphism of locally ringed spaces if for every point the induced stalk map
is a local ring homomorphism, meaning that the maximal ideal of maps into the maximal ideal of .
Depends on
Used by
- Minimal tangent dimension and homogeneous regularity Corollary
- A morphism of ringed spaces need not be a morphism of locally ringed spaces Counterexample
- Analytic topology and finite smooth algebraic stratifications of complex varieties Definition
- Classical algebraic prevarieties, regular maps, and varieties Definition
- G-linearizations of invertible sheaves on a complex G-variety Definition
- Good and geometric quotients for group actions Definition
- Morphisms of schemes Definition
- The map of affine spectra induced by a ring homomorphism Definition
- A local morphism of stalks induces a residue-field map Lemma
- A principal localization identifies its spectrum with a distinguished open Lemma
- Differentials, open restriction, and the chain rule Lemma
- Pushouts of closed immersions exist Lemma
- The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
- The stalk maps induced by a ring map are local Lemma
- Compatible open pieces of ringed or locally ringed spaces glue 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
- The Stacks Project, Definition 26.2.1 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Section 6.3.1 (standard reference, not scraped)