affine scheme 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 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 (see below) 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. See at functorial geometry.

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

Isbell duality

(e.g. Hartschorne 77, chapter II, prop. 2.3)

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, Springer 1977

On an approach on schemes as locally representable sheaves on the site of affine schemes see

• Demazure, P. Gabriel, Algebraic groups

• Daniel Murfet, Schemes via noncommutative localisation (2005) pdf

In this note we give an exposition of the well-known results of Gabriel, which show how to define affine schemes in terms of the theory of noncommutative localisation.

For affine schemes in cubical type theory, see:

• Anders Mörtberg, Max Zeuner, A Univalent Formalization of Affine Schemes (arXiv:2212.02902)

• Max Zeuner, A univalent formalization of affine schemes, 20 October 2022 (slides, video)