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affine scheme

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 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 (see below) says that there is a bijection

CRing(R,Ξ“ Yπ’ͺ Y)β‰…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(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. See at functorial geometry.

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

Remark

There is no similar equation the other way round, that is β€œRing(Ξ“ Yπ’ͺ Y,R)β‰…Scheme(SpecR,Y)Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)”. As a mnemonic, note that with ordinary Galois connections between power sets, one is always homming into (not out of)? the functorial construction. More geometrically, consider the example Y=β„™ nY = \mathbb{P}^n and R=β„€R = \mathbb{Z}. Then the left hand side consists of all the β„€\mathbb{Z}-valued points of β„™ n\mathbb{P}^n (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism β„€β†’β„€\mathbb{Z} \to \mathbb{Z}, since π’ͺ β„™ n(β„™ n)β‰…β„€\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}.

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

Isbell duality

Proposition

(affine schemes form full subcategory of opposite of rings)

The functor

π’ͺ:Schemes Aff⟢Ring op \mathcal{O} \;\colon\; Schemes_{Aff} \longrightarrow Ring^{op}

from affine schemes to their global rings of functions is a fully faithful functor.

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

Remark

(Isbell duality between geometry and algebra)

Prop. is the analog in algebraic geometry of similar statements of Isbell duality between geometry and algebra, such as Gelfand duality or Milnor's exercise.

Isbell 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}β†ͺGelfand-KolmogorovAlg ℝ op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">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}β†ͺalmost 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}β†ͺMilnor'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}β†ͺMilnor's exercise Alg β„€ 2AAAA op ↦ C ∞(ℝ n)βŠ—βˆ§ ‒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 𝔀A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}Aβ†ͺALada-MarklA sdgcAlg op ↦ CE(𝔀)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} (β€œFDAs”)

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 observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}SchrΓΆdinger 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 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, Springer 1977

  • Demazure, Gabriel, Algebraic groups

Last revised on July 26, 2018 at 06:53:57. See the history of this page for a list of all contributions to it.