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 map of affine spectra induced by a ring homomorphism
Definition
A homomorphism gives the continuous contraction map , . On its sheaf map is the localization map ; these commute with restrictions and define a morphism of ringed spaces. Its localness is verified next.
Depends on
Used by
- Flat and finite type is not open without finite presentation Counterexample
- An affine nilpotent thickening Definition
- The coordinate Hopf algebra of an affine group scheme Definition
- Two-affine projective line and its twists Definition
- Incidence projection has closed determinantal image Example
- Pulling a divisor back along the cusp normalization Example
- The spectrum of a product ring is a disjoint union Example
- A dominant map has a surjective differential on a dense source open Lemma
- A quasi-compact image stable under specialization is closed Lemma
- A quasi-compact immersion with nonclosed image has a boundary specialization Lemma
- A surjective ring map induces a closed immersion of affine spectra Lemma
- Affine pushforward algebra localizes Lemma
- An associated sheaf restricts to an associated sheaf on an affine open Lemma
- Closed gluing of two projective three-spaces is proper Lemma
- Constructible subsets stable under generalisation are open in an affine spectrum Lemma
- Fpqc covers are universally submersive Lemma
- Scheme pullback preserves quasi-coherence Lemma
- Sections of a sheaf flat over the base are flat over affine opens 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
- Affine schemes are contravariantly equivalent to commutative rings Theorem
- Quasi-coherence of pushforward for qcqs morphisms Theorem
Dependency tree · two levels
15 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, Section 26.6 (standard reference, not scraped)