Affine schemes are contravariantly equivalent to commutative rings

Depends on

• Affine schemes and their coordinate rings
• The map of affine spectra induced by a ring homomorphism
• The stalk maps induced by a ring map are local
• Global functions on Spec A recover A
• Sections and restrictions on distinguished opens of an affine scheme

Used by

• Affine-scheme isomorphisms are exactly coordinate-ring isomorphisms in reverse direction Corollary
• Maps from proper integral schemes to the affine line have closed-point image Corollary
• A nonradical ideal need not enlarge every tangent space Counterexample
• Equal fibre points can have different multiplicities Counterexample
• Rational points do not detect the group-scheme structure of alphaₚ and muₚ Counterexample
• The equation must define the intended scheme Counterexample
• The orbit set of k-points need not be the k-points of the fppf quotient sheaf Counterexample
• Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
• Affine-local quasi-coherent algebras before general sheaf theory Definition
• The coordinate Hopf algebra of an affine group scheme Definition
• The functor of points of an affine scheme Definition
• A node has two distinct tangent directions Example
• A nondegenerate projective quadric Example
• Additive and infinitesimal group schemes Example
• An irreducible curve can have arbitrarily large tangent dimension Example
• Classical bilinear-form equations linearize to matrix spaces Example
• Different singularities can share a tangent cone Example
• Dual numbers compute the tangent spaces of the general and special linear groups Example
• Finite power map of the affine line Example
• Incidence projection has closed determinantal image Example
• The cusp retains a doubled tangent line Example
• The group schemes Ga, Gm, and GLn Example
• The Hopf algebra of a split torus and its root-of-unity subgroups Example
• The product of two parabolas: a block Jacobian and the direct-sum formula Example
• The tangent space of the parabola at a general point and the local parameter Example
• Three axes are not three coplanar lines Example
• Valuative extension of projective coordinates Example
• A finite-type affine algebraic group has a faithful rational representation Lemma
• A quasi-compact image stable under specialization is closed Lemma
• A tangent direction is realized by a local smooth curve Lemma
• A transverse hyperplane slice is smooth at the chosen point Lemma
• Affine pushforward algebra localizes Lemma
• Affine-local flatness Lemma
• Affineness and finiteness of morphisms descend under fppf base change Lemma
• An associated sheaf restricts to an associated sheaf on an affine open Lemma
• Birational morphisms restrict to isomorphisms between principal affine opens Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Critical loci have small images in characteristic zero Lemma
• Faithfully flat descent of modules and affine algebras is effective Lemma

…and 36 more results.

Dependency tree · two levels

14 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, Lemmas 26.6.4 and 26.6.5 (standard reference, not scraped)