Sheaf of algebras

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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

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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

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  • 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

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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

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References

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  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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