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 -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 from the category of quasi-coherent (sheaves of) -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 to [1]

Affine morphism

A morphism of schemes is called affine if has an open affine cover 's such that 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 be an affine morphism between schemes and a locally ringed space together with a map . Then the natural map between the sets:

is bijective.[4]

Examples

  • Let be the normalization of an algebraic variety X. Then, since f is finite, is quasi-coherent and .
  • Let be a locally free sheaf of finite rank on a scheme X. Then is a quasi-coherent -algebra and is the associated vector bundle over X (called the total space of .)
  • More generally, if F is a coherent sheaf on X, then one still has , 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 of pairs consisting of a ringed space morphism and an -module . Then the formation of direct images determines the contravariant functor from to the category of pairs consisting of an -algebra A and an A-module M that sends each pair to the pair .

Now assume S is a scheme and then let be the subcategory consisting of pairs such that is an affine morphism between schemes and a quasi-coherent sheaf on . Then the above functor determines the equivalence between and the category of pairs consisting of an -algebra A and a quasi-coherent -module .[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 -algebra and then take its global Spec: . Then, for each quasi-coherent A-module M, there is a corresponding quasi-coherent -module such that called the sheaf associated to M. Put in another way, determines an equivalence between the category of quasi-coherent -modules and the quasi-coherent -modules.

See also

References

  1. EGA 1971, Ch. I, Théorème 9.1.4.
  2. EGA 1971, Ch. I, Definition 9.1.1.
  3. Stacks Project, Tag 01S5.
  4. EGA 1971, Ch. I, Proposition 9.1.5.
  5. EGA 1971, Ch. I, Théorème 9.2.1.
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