Grothendieck topology: Difference between revisions
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imported>Giovanni Antonio DiMatteo No edit summary |
imported>Aleksander Stos m (Grothendieck Topology moved to Grothendieck topology: convention) |
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Revision as of 11:52, 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)