Grothendieck topology: Difference between revisions
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imported>Giovanni Antonio DiMatteo (New page: 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 shea...) |
imported>Giovanni Antonio DiMatteo No edit summary |
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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) | ||
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Revision as of 15:24, 2 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)
