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.
Global functions on Spec A recover A
Statement
The canonical map is an isomorphism, including when .
Facts & Assumptions
Given: The basic-open section calculation.
Proof
.
The basic-open calculation gives .
The canonical map is an isomorphism, also for .
Depends on
Used by
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- Maps from proper integral schemes to the affine line have closed-point image Corollary
- Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
- Affine schemes and their coordinate rings Definition
- Sheaf total quotient rings Definition
- The coordinate Hopf algebra of an affine group scheme Definition
- Projective zero-space over an affine base Example
- The spectrum of a field is a one-point affine scheme Example
- The zero ring has empty spectrum Example
- A proper geometrically integral affine scheme is a point Lemma
- Affine charts recover the algebraic module of differentials Lemma
- Affineness of a field form of a diagonalizable group Lemma
- Closed gluing of two projective three-spaces is proper Lemma
- Global sections commute with extension of scalars over a field Lemma
- Power extension over a normal affine domain Lemma
- Regular hyperplane step for coherent support induction Lemma
- The sheaf attached to a module on an affine scheme Lemma
- The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
- The standard open D_+(f) of a projective quotient is the affine chart Spec((S_f)₀) Lemma
- Affine group schemes of finite type are antiequivalent to finitely generated commutative Hopf algebras Theorem
- Affine schemes are contravariantly equivalent to commutative rings Theorem
- Morphisms to an affine scheme and global sections Theorem
- Representability and smoothness of concentrator subschemes Theorem
Dependency tree · two levels
4 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 26.5.4 (standard reference, not scraped)