affine scheme

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 locally 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** 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 $Aff$.

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

There is no similar equation the other way round, that is β$Ring(\Gamma_Y\mathcal{O}_Y, R) \cong Scheme(Spec R, Y)$β. 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 $\mathcal{O}_{\mathbb{P}^n}(\mathbb{P}^n) \cong \mathbb{Z}$.

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 βrelativeβ 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.

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 $\mathcal{F}\mapsto\mathcal{F}(Spec R)$.

- Robin Hartshorne,
*Algebraic geometry* - Demazure, Gabriel,
*Algebraic groups*

category: algebraic geometry

Revised on May 5, 2015 21:04:38
by Ingo Blechschmidt
(95.90.194.49)