nLab affine scheme

Definition

General

An affine scheme is a scheme that as a sheaf on the opposite category CRing${}^{op}$ 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 $Aff\hookrightarrow Scheme$ of the category of schemes.

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

$$CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).$$

In other words, for fixed $Y$, and for varying $R$ there is a restricted functor

$$Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$$

and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff$ is fully faithful. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$. 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 “$Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)$”. 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 $Y = \mathbb{P}^n$ and $R = \mathbb{Z}$. Then the left hand side consists of all the $\mathbb{Z}$-valued points of $\mathbb{P}^n$ (of which there are many). On the other hand, the right hand side only contains the unique ring homomorphism $\mathbb{Z} \to \mathbb{Z}$, since $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.

Relative affine schemes

A relative affine scheme over a scheme $Y$ is a relative scheme $f:X\to Y$ isomorphic to the spectrum of a (commutative unital) algebra $A$ in the category of quasicoherent $\mathcal{O}_Y$-modules; such a “relative” spectrum has been introduced by Grothendieck. It is characterized by the property that for every open $V\subset Y$ the inverse image $f^{-1}V\subset X$ is an open affine subscheme of $X$ isomorphic to $Spec(A(V))$ and such open affines glue in such a way that $f^{-1}V\hookrightarrow f^{-1}W$ corresponds to the restriction morphism $A(W)\to A(V)$ 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

$$\mathcal{O} \;\colon\; Schemes_{Aff} \longrightarrow Ring^{op}$$

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.

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\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}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\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 }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

Affine Serre’s theorem

Affine Serre's theorem

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

Related concepts

• 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)

category: algebraic geometry