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

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

is bijective.

Examples

 * Let $$f: \widetilde{X} \to X$$ be the normalization of an algebraic variety X. Then, since f is finite, $$f_* \mathcal{O}_{\widetilde{X}}$$ is quasi-coherent and $$\operatorname{Spec}_X(f_* \mathcal{O}_{\widetilde{X}}) = \widetilde{X}$$.
 * Let $$E$$ be a locally free sheaf of finite rank on a scheme X. Then $$\operatorname{Sym}(E^*)$$ is a quasi-coherent $$\mathcal{O}_X$$-algebra and $$\operatorname{Spec}_X(\operatorname{Sym}(E^*)) \to X$$ 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 $$\operatorname{Spec}_X(\operatorname{Sym}(F)) \to X$$, 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 $$\mathcal{O}_X$$-module $$M$$. Then the formation of direct images determines the contravariant functor from $$C_S$$ to the category of pairs consisting of an $$\mathcal{O}_S$$-algebra A and an A-module M that sends each pair $$(f, M)$$ to the pair $$(f_* \mathcal{O}, f_* M)$$.

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