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 [math]\displaystyle{ \mathcal{O} X }[/math]-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 [math]\displaystyle{ \operatorname{Spec}_X }[/math] from the category of quasi-coherent (sheaves of) [math]\displaystyle{ \mathcal{O}_X }[/math]-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 [math]\displaystyle{ f: Y \to X }[/math] to [math]\displaystyle{ f_* \mathcal{O}_Y. }[/math]

Affine morphism

A morphism of schemes [math]\displaystyle{ f: X \to Y }[/math] is called affine if [math]\displaystyle{ Y }[/math] has an open affine cover [math]\displaystyle{ U_i }[/math]'s such that [math]\displaystyle{ f^{-1}(U_i) }[/math] 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 [math]\displaystyle{ f: X \to Y }[/math] be an affine morphism between schemes and [math]\displaystyle{ E }[/math] a locally ringed space together with a map [math]\displaystyle{ g: E \to Y }[/math]. Then the natural map between the sets:

[math]\displaystyle{ \operatorname{Mor}_Y(E, X) \to \operatorname{Hom}_{\mathcal{O}_Y-\text{alg}}(f_* \mathcal{O}_X, g_* \mathcal{O}_E) }[/math]

is bijective.

Examples

• Let [math]\displaystyle{ f: \widetilde{X} \to X }[/math] be the normalization of an algebraic variety X. Then, since f is finite, [math]\displaystyle{ f_* \mathcal{O}_{\widetilde{X}} }[/math] is quasi-coherent and [math]\displaystyle{ \operatorname{Spec}_X(f_* \mathcal{O}_{\widetilde{X}}) = \widetilde{X} }[/math].
• Let [math]\displaystyle{ E }[/math] be a locally free sheaf of finite rank on a scheme X. Then [math]\displaystyle{ \operatorname{Sym}(E^*) }[/math] is a quasi-coherent [math]\displaystyle{ \mathcal{O}_X }[/math]-algebra and [math]\displaystyle{ \operatorname{Spec}_X(\operatorname{Sym}(E^*)) \to X }[/math] is the associated vector bundle over X (called the total space of [math]\displaystyle{ E }[/math].)
• More generally, if F is a coherent sheaf on X, then one still has [math]\displaystyle{ \operatorname{Spec}_X(\operatorname{Sym}(F)) \to X }[/math], 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 [math]\displaystyle{ C_S }[/math] of pairs [math]\displaystyle{ (f, M) }[/math] consisting of a ringed space morphism [math]\displaystyle{ f: X \to S }[/math] and an [math]\displaystyle{ \mathcal{O}_X }[/math]-module [math]\displaystyle{ M }[/math]. Then the formation of direct images determines the contravariant functor from [math]\displaystyle{ C_S }[/math] to the category of pairs consisting of an [math]\displaystyle{ \mathcal{O}_S }[/math]-algebra A and an A-module M that sends each pair [math]\displaystyle{ (f, M) }[/math] to the pair [math]\displaystyle{ (f_* \mathcal{O}, f_* M) }[/math].

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

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 [math]\displaystyle{ \mathcal{O}_S }[/math]-algebra and then take its global Spec: [math]\displaystyle{ f: X = \operatorname{Spec}_S(A) \to S }[/math]. Then, for each quasi-coherent A-module M, there is a corresponding quasi-coherent [math]\displaystyle{ \mathcal{O}_X }[/math]-module [math]\displaystyle{ \widetilde{M} }[/math] such that [math]\displaystyle{ f_* \widetilde{M} \simeq M, }[/math] called the sheaf associated to M. Put in another way, [math]\displaystyle{ f_* }[/math] determines an equivalence between the category of quasi-coherent [math]\displaystyle{ \mathcal{O}_X }[/math]-modules and the quasi-coherent [math]\displaystyle{ A }[/math]-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