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affine scheme

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Definition

General

An affine scheme is a scheme that as a sheaf on the opposite category CRing op{}^{op} of commutative rings (or equivalently as a sheaf on the subcategory of finitely presented rings) is representable. In a ringed space picture an affine scheme is a locally ringed space which is locally isomorphic to the prime spectrum of a commutative ring. Affine schemes form a full subcategory Affβ†ͺSchemeAff\hookrightarrow Scheme of the category of schemes.

The correspondence Y↦Spec(Ξ“ Yπ’ͺ Y)Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y) extends to a functor Schemeβ†’AffScheme\to Aff. The fundamental theorem on morphisms of schemes says that there is a bijection

CRing(Ξ“ Yπ’ͺ Y,R)β‰…Scheme(SpecR,Y). CRing(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y).

In other words, for fixed YY, and for varying RR there is a restricted functor

Scheme(βˆ’,Y)| Aff op=h Y| Aff op=h Y| CRing:CRingβ†’Set, Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,

and the functor Y↦h Y| CRingY\mapsto h_Y|_{CRing} from schemes to presheaves on AffAff is fully faithful. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on AffAff.

There is an analogue of this theorem for relative noncommutative schemes in the sense of Rosenberg.

Relative affine schemes

A relative affine scheme over a scheme YY is a relative scheme f:Xβ†’Yf:X\to Y isomorphic to the spectrum of a (commutative unital) algebra AA in the category of quasicoherent π’ͺ Y\mathcal{O}_Y-modules; such a β€œrelative” spectrum has been introduced by Grothendieck. It is characterized by the property that for every open VβŠ‚YV\subset Y the inverse image f βˆ’1VβŠ‚Xf^{-1}V\subset X is an open affine subscheme of XX isomorphic to Spec(A(V))Spec(A(V)) and such open affines glue in such a way that f βˆ’1Vβ†ͺf βˆ’1Wf^{-1}V\hookrightarrow f^{-1}W corresponds to the restriction morphism A(W)β†’A(V)A(W)\to A(V) of algebras.

Relative affine scheme is a concrete way to represent an affine morphism of schemes.

Properties

Affine Serre’s theorem

Affine Serre's theorem

Given a commutative unital ring RR there is an equivalence of categories RModβ†’Qcoh(SpecR){}_R Mod\to Qcoh(Spec R) between the category of RR-modules and the category of quasicoherent sheaves of π’ͺ SpecR\mathcal{O}_{Spec R}-modules given on objects by M↦M˜M\mapsto \tilde{M} where M˜\tilde{M} is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization M˜(D f)=R[f βˆ’1]βŠ— RM\tilde{M}(D_f) = R[f^{-1}]\otimes_R M where D fD_f is the principal Zariski open set underlying SpecR[f βˆ’1]βŠ‚SpecRSpec R[f^{-1}]\subset Spec R, and the restrictions are given by the canonical maps among the localizations. The action of π’ͺ SpecR\mathcal{O}_{Spec R} is defined using a similar description of π’ͺ SpecR=R˜\mathcal{O}_{Spec R} = \tilde{R}. Its right adjoint (quasi)inverse functor is given by the global sections functor ℱ↦ℱ(SpecR)\mathcal{F}\mapsto\mathcal{F}(Spec R).

References

  • Robin Hartshorne, Algebraic geometry
  • Demazure, Gabriel, Algebraic groups

Revised on November 21, 2013 00:48:29 by Urs Schreiber (82.169.114.243)