affine scheme (Rev #11) in nLab

Contents

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 (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)

duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}	A\phantom{A}categoryA\phantom{A}	A\phantom{A}dual categoryA\phantom{A}	A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}	A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}	A\phantom{A}↪Gelfand-KolmogorovAlg ℝ op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}	A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}	A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}	A\phantom{A}≃Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}	A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}	A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}	A\phantom{A}≔Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}	A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}	A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}	A\phantom{A}↪almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}	A\phantom{A}fin. gen.A\phantom{A} A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A} A\phantom{A}geometryA\phantom{A}	A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}	A\phantom{A}≔Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}	A\phantom{A}fin. gen. A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}	A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}	A\phantom{A}↪Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}	A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}	A\phantom{A}SuperSpaces Cart ℝ n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}	A\phantom{A}↪Milnor's exercise Alg ℤ 2AAAA op ↦ C ∞(ℝ n)⊗∧ •ℝ q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}	A\phantom{A}supercommutativeA\phantom{A} A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A} A\phantom{A}supergeometryA\phantom{A} A\phantom{A}(super Lie theory)A\phantom{A}	ASuperL ∞Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}	A↪ALada-MarklA sdgcAlg op ↦ CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}	A\phantom{A}differential graded-commutativeA\phantom{A} A\phantom{A}superalgebra A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}algebraA\phantom{A}	A\phantom{A}geometryA\phantom{A}
A\phantom{A}Poisson algebraA\phantom{A}	A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}	A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observables	A\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg picture	A\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}AQFTA\phantom{A}	A\phantom{A}FQFTA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}	A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}	A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}	A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}	A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}	A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}	A\phantom{A}cobordism representationA\phantom{A}

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

• Demazure, Gabriel, Algebraic groups