“Sheaf of algebras”的意思、由来-开放百科全书

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Sheaf of algebras

释义

Affine morphism
Examples
The formation of direct images
See also
References
External links

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 Sepc: . 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.

References

1. ^{{harvnb|EGA|1971|loc=Ch. I, Théorème 9.1.4.}}
2. ^{{harvnb|EGA|1971|loc=Ch. I, Definition 9.1.1.}}
3. ^{{Citation | title=Stacks Project, Tag 01S5 | url=https://stacks.math.columbia.edu/tag/01S5}}.
4. ^{{harvnb|EGA|1971|loc=Ch. I, Proposition 9.1.5.}}
5. ^{{harvnb|EGA|1971|loc=Ch. I, Théorème 9.2.1.}}

{{EGA|book=1-2}}
{{Hartshorne AG}}

External links

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

2 : Sheaf theory|Morphisms of schemes

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Affine morphism

Examples

The formation of direct images

See also

References

External links