Grothendieck topology: Difference between revisions
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The notion of a ''Grothendieck topology'' or ''site'' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin. | {{subpages}} | ||
The notion of a '''Grothendieck topology''' or '''site'''' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin. | |||
==Definition== | ==Definition== | ||
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#A standard topological space <math>X</math> becomes a category <math>op(X)</math> when you regard the open subsets of <math>X</math> as objects, and morphisms are inclusions. An open covering of open subsets <math>U</math> clearly verify the axioms above for coverings in a site. Notice that a [[presheaf]] of rings is just a contravariant functor from the category <math>op(X)</math> into the category of rings. | #A standard topological space <math>X</math> becomes a category <math>op(X)</math> when you regard the open subsets of <math>X</math> as objects, and morphisms are inclusions. An open covering of open subsets <math>U</math> clearly verify the axioms above for coverings in a site. Notice that a [[presheaf]] of rings is just a contravariant functor from the category <math>op(X)</math> into the category of rings. | ||
#'''The Small Étale Site''' Let <math>S</math> be a scheme. Then the [[category of étale schemes]] over <math>S</math> (i.e., <math>S</math>-schemes <math>X</math> over <math>S</math> whose structural morphisms are étale) | #'''The Small Étale Site''' Let <math>S</math> be a scheme. Then the [[Étale morphism|category of étale schemes]] over <math>S</math> (i.e., <math>S</math>-schemes <math>X</math> over <math>S</math> whose structural morphisms are étale) becomes a site if we require that coverings are jointly surjective; that is, | ||
==Sheaves on Sites== | ==Sheaves on Sites== | ||
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In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor | In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor | ||
such that for all coverings <math>\{U_i\to U\}\in cov(T)</math>, the diagram | such that for all coverings <math>\{U_i\to U\}\in cov(T)</math>, the diagram | ||
<math>0\to F(U)\to \ | <math>0\to F(U)\to \prod F(U_i)\to \prod F(U_i\times_U U_j)</math> | ||
is exact. | is exact.[[Category:Suggestion Bot Tag]] | ||
[[Category: |
Latest revision as of 06:00, 24 August 2024
The notion of a Grothendieck topology or site' captures the essential properties necessary for constructing a robust theory of cohomology of sheaves. The theory of Grothendieck topologies was developed by Alexander Grothendieck and Michael Artin.
Definition
A Grothendieck topology consists of
- A category, denoted
- A set of coverings , denoted , such that
- for each object of
- If , and is any morphism in , then the canonical morphisms of the fiber products determine a covering
- If and , then
Examples
- A standard topological space becomes a category when you regard the open subsets of as objects, and morphisms are inclusions. An open covering of open subsets clearly verify the axioms above for coverings in a site. Notice that a presheaf of rings is just a contravariant functor from the category into the category of rings.
- The Small Étale Site Let be a scheme. Then the category of étale schemes over (i.e., -schemes over whose structural morphisms are étale) becomes a site if we require that coverings are jointly surjective; that is,
Sheaves on Sites
In analogy with the situation for topological spaces, a presheaf may be defined as a contravariant functor such that for all coverings , the diagram is exact.