affine scheme (Rev #9, changes) in nLab

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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 Aff↪SchemeAff\hookrightarrow Scheme of the category of schemes.

The correspondence Y↦Spec(Γ Y𝒪 Y)Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y) extends to a functor Scheme→AffScheme\to Aff. The fundamental theorem on morphisms of schemes says that there is a bijection

CRing(R,Γ Y𝒪 Y)≅Scheme(Y,SpecR). CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).

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

Scheme(−,Y)| Aff op=h Y| Aff op=h Y| CRing:CRing→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:X→Yf:X\to Y isomorphic to the spectrum of a (commutative unital) algebra AA in the category of quasicoherent 𝒪 Y\mathcal{O}_Y-modules; such a “relative” spectrum has been introduced by Grothendieck. It is characterized by the property that for every open V⊂YV\subset Y the inverse image f −1V⊂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 −1V↪f −1Wf^{-1}V\hookrightarrow f^{-1}W corresponds to the restriction morphism A(W)→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 RMod→Qcoh(SpecR){}_R Mod\to Qcoh(Spec R) between the category of RR-modules and the category of quasicoherent sheaves of 𝒪 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 −1]⊗ RM\tilde{M}(D_f) = R[f^{-1}]\otimes_R M where D fD_f is the principal Zariski open set underlying SpecR[f −1]⊂SpecRSpec R[f^{-1}]\subset Spec R, and the restrictions are given by the canonical maps among the localizations. The action of 𝒪 SpecR\mathcal{O}_{Spec R} is defined using a similar description of 𝒪 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).

• spectral topological space

References

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