Affine Morphism

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 $\mathcal_X$-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 $\operatorname_X$ from the category of quasi-coherent (sheaves of) $\mathcal_X$-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 $f_* \mathcal_Y.$

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 $f^(U_i)$ 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:
:$\operatorname_Y(E, X) \to \operatorname_(f_* \mathcal_X, g_* \mathcal_E)$
is bijective.

Examples

*Let $f: \widetilde \to X$ be the normalization of an algebraic variety ''X''. Then, since ''f'' is finite, $f_* \mathcal_$ is quasi-coherent and $\operatorname_X(f_* \mathcal_) = \widetilde$.
*Let $E$ be a locally free sheaf of finite rank on a scheme ''X''. Then $\operatorname(E^*)$ is a quasi-coherent $\mathcal_X$-algebra and $\operatorname_X(\operatorname(E^*)) \to X$ 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 $\operatorname_X(\operatorname(F)) \to X$, usually called the abelian hull of ''F''; see Cone (algebraic geometry)#Examples.

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 $\mathcal_X$-module $M$. Then the formation of direct images determines the contravariant functor from $C_S$ to the category of pairs consisting of an $\mathcal_S$-algebra ''A'' and an ''A''-module ''M'' that sends each pair $(f, M)$ to the pair $(f_* \mathcal, f_* M)$.

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

See also

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

References

*
*{{Hartshorne AG

External links

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

Sheaf theory
Morphisms of schemes