Grothendieck topology: Difference between revisions
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imported>Aleksander Stos m (Grothendieck Topology moved to Grothendieck topology: convention) |
imported>Giovanni Antonio DiMatteo (→Examples: send link to yet-to-be written page on etale cohomology and etale site) |
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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 [[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) | ||
[[Category:CZ Live]] | [[Category:CZ Live]] | ||
[[Category:Mathematics Workgroup]] | [[Category:Mathematics Workgroup]] | ||
[[Category:Stub Articles]] | [[Category:Stub Articles]] | ||
Revision as of 13:04, 4 December 2007
The notion of a Grothendieck topology or site is a category which has the features of open covers in topological spaces necessary for generalizing much of sheaf cohomology to sheaves on more general sites.
Definition
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)
