Mathematics Colloquia and Seminars

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In this talk I will provide the motivation, basic definitions, and some examples of Grothendieck topologies and Weil cohomology theories. In algebraic geometry the Zariski topology is excellent for encoding sheaves, but it is also exceptionally coarse. This makes the corresponding cohomology theory poorly behaved. For example, if we take a constant sheaf on any variety, the corresponding cohomology has trivial higher cohomology, even though the associated topological space may not be trivial. If we are working over C we can repair this by using the analytic topology and using de Rham or singular cohomology. These fulfill some useful axioms like Poincare duality on smooth varieties, which make them examples of what are called Weil cohomology theories. It's useful to create similar cohomology theories in characteristic p, and so two other examples of Weil Cohomology: l-adic and Crystalline cohomology, were developed. To do this, the open sets (ie open immersions) of Zariski topology are replaced by different kinds of schemes and maps. The new "open sets" may no longer be subsets of our space! For example Etale "open sets" can be thought of open subsets of unbranched covers. These give us the so called Grothendieck topologies.