Sheaf of algebras

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 ${\displaystyle {\mathcal {O}}_{X}}$-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 ${\displaystyle \operatorname {Spec} _{X}}$ from the category of quasi-coherent (sheaves of) ${\displaystyle {\mathcal {O}}_{X}}$-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 ${\displaystyle f:Y\to X}$ to ${\displaystyle f_{*}{\mathcal {O}}_{Y}.}$^[1]

A morphism of schemes ${\displaystyle f:X\to Y}$ is called affine if ${\displaystyle Y}$ has an open affine cover ${\displaystyle U_{i}}$'s such that ${\displaystyle f^{-1}(U_{i})}$ are affine.^[2] 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.^[3]

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

${\displaystyle \operatorname {Mor} _{Y}(E,X)\to \operatorname {Hom} _{{\mathcal {O}}_{Y}-{\text{alg}}}(f_{*}{\mathcal {O}}_{X},g_{*}{\mathcal {O}}_{E})}$

is bijective.^[4]

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

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

Now assume S is a scheme and then let ${\displaystyle \operatorname {Aff} _{S}\subset C_{S}}$ be the subcategory consisting of pairs ${\displaystyle (f:X\to S,M)}$ such that ${\displaystyle f}$ is an affine morphism between schemes and ${\displaystyle M}$ a quasi-coherent sheaf on ${\displaystyle X}$. Then the above functor determines the equivalence between ${\displaystyle \operatorname {Aff} _{S}}$ and the category of pairs ${\displaystyle (A,M)}$ consisting of an ${\displaystyle {\mathcal {O}}_{S}}$-algebra A and a quasi-coherent ${\displaystyle A}$-module ${\displaystyle M}$.^[5]

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

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

• Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). Vol. 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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