Sheaf of algebras

Short description: Type of ringed space

In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of ${𝒪}X$-modules. It is quasi-coherent if it is so as a module.

When X is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor ${\mathrm{Spec}}_X$ from the category of quasi-coherent (sheaves of) ${{𝒪}}_X$-algebras on X to the category of schemes that are affine over X (defined below). Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism $f:Y\to X$ to $f_{*}{{𝒪}}_Y.$

Affine morphism

A morphism of schemes $f:X\to Y$ is called affine if $Y$ has an open affine cover $U_i$'s such that $f^{-1}(U_i)$ are affine. For example, a finite morphism is affine. An affine morphism is quasi-compact and separated; in particular, the direct image of a quasi-coherent sheaf along an affine morphism is quasi-coherent.

The base change of an affine morphism is affine.

Let $f:X\to Y$ be an affine morphism between schemes and $E$ a locally ringed space together with a map $g:E\to Y$. Then the natural map between the sets:

${\mathrm{Mor}}_Y(E,X)\to {\mathrm{Hom}}_{{{𝒪}}_Y-\text{alg}}(f_{*}{{𝒪}}_X,g_{*}{{𝒪}}_E)$

is bijective.

Examples

• Let $f:\tilde{X}\to X$ be the normalization of an algebraic variety X. Then, since f is finite, $f_{*}{{𝒪}}_{\tilde{X}}$ is quasi-coherent and ${\mathrm{Spec}}_X(f_{*}{{𝒪}}_{\tilde{X}})=\tilde{X}$.
• Let $E$ be a locally free sheaf of finite rank on a scheme X. Then $\mathrm{Sym}(E^{*})$ is a quasi-coherent ${{𝒪}}_X$-algebra and ${\mathrm{Spec}}_X(\mathrm{Sym}(E^{*}))\to X$ is the associated vector bundle over X (called the total space of $E$.)
• More generally, if F is a coherent sheaf on X, then one still has ${\mathrm{Spec}}_X(\mathrm{Sym}(F))\to X$, usually called the abelian hull of F; see Cone (algebraic geometry).

The formation of direct images

Given a ringed space S, there is the category $C_S$ of pairs $(f,M)$ consisting of a ringed space morphism $f:X\to S$ and an ${{𝒪}}_X$-module $M$. Then the formation of direct images determines the contravariant functor from $C_S$ to the category of pairs consisting of an ${{𝒪}}_S$-algebra A and an A-module M that sends each pair $(f,M)$ to the pair $(f_{*}{𝒪},f_{*}M)$.

Now assume S is a scheme and then let ${\mathrm{Aff}}_S\subset C_S$ be the subcategory consisting of pairs $(f:X\to S,M)$ such that $f$ is an affine morphism between schemes and $M$ a quasi-coherent sheaf on $X$. Then the above functor determines the equivalence between ${\mathrm{Aff}}_S$ and the category of pairs $(A,M)$ consisting of an ${{𝒪}}_S$-algebra A and a quasi-coherent $A$-module $M$.

The above equivalence can be used (among other things) to do the following construction. As before, given a scheme S, let A be a quasi-coherent ${{𝒪}}_S$-algebra and then take its global Spec: $f:X={\mathrm{Spec}}_S(A)\to S$. Then, for each quasi-coherent A-module M, there is a corresponding quasi-coherent ${{𝒪}}_X$-module $\tilde{M}$ such that $f_{*}\tilde{M}\simeq M,$ called the sheaf associated to M. Put in another way, $f_{*}$ determines an equivalence between the category of quasi-coherent ${{𝒪}}_X$-modules and the quasi-coherent $A$-modules.

See also

• quasi-affine morphism
• Serre's theorem on affineness

References

• Grothendieck, Alexandre; Dieudonné, Jean (1971) (in French). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften. 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9 

External links

• https://ncatlab.org/nlab/show/affine+morphism