Affine schemes and their coordinate rings

Depends on

• The underlying space of an affine spectrum
• Spec A with its structure sheaf is a locally ringed space
• Global functions on Spec A recover A

Used by

• Every affine scheme is quasi-compact Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite type need not mean finite presentation Counterexample
• The fixed point theorem fails without completeness: the additive group acts on the affine line by translations Counterexample
• Affine open subschemes Definition
• An affine nilpotent thickening Definition
• Borel subgroups, maximal tori and Borel pairs Definition
• Closed points of an affine scheme Definition
• Crossed homomorphisms, principal crossed homomorphisms and Hochschild extensions Definition
• Degree divisor proper curve Definition
• Generic points of irreducible closed subsets Definition
• Limits of one-parameter orbits and concentrator subschemes Definition
• Locally factorial scheme Definition
• Parabolic subgroups of an affine algebraic group Definition
• Projective scheme of a homogeneous quotient and its standard affine charts Definition
• Reduced affine schemes Definition
• Relative projective space from standard charts Definition
• Schemes Definition
• Sheaf total quotient rings Definition
• The affine scheme of dual numbers Definition
• The coordinate Hopf algebra of an affine group scheme Definition
• The cotangent complex of a morphism of schemes Definition
• The functor of points of an affine scheme Definition
• The map of affine spectra induced by a ring homomorphism Definition
• The scheme-theoretic tangent cone at a point Definition
• Trigonalizable algebraic groups Definition
• Two-affine projective line and its twists Definition
• Unipotent algebraic groups and unipotent representations Definition
• Weil divisor normal noetherian scheme Definition
• Divisor of a rational function on the projective line Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• The affine line is an algebraic space Example
• The empty morphism is finite, proper and projective Example
• The spectrum of a product ring is a disjoint union Example
• A Borel subgroup of maximal dimension is the stabilizer of a maximal flag Lemma
• A meromorphic unit has locally finite nonzero order support Lemma
• A surjective ring map induces a closed immersion of affine spectra Lemma
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma

…and 33 more results.

Dependency tree · two levels

10 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 26.5.5 (standard reference, not scraped)