In algebraic geometry, the '''smooth topology''' is a certain Grothendieck topology which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in, say, the étale sheaf <math>\mathbb{Q}_l</math>.
To understand the problem that motivates the notion, consider the classifying stack <math>B\mathbb{G}_m</math> over <math>\operatorname{Spec} \mathbf{F}_q</math>. Then <math>B\mathbb{G}_m = \operatorname{Spec} \mathbf{F}_q</math> in the étale topology;<ref>{{harvnb|Behrend|2003|loc=Proposition 5.2.9; in particular, the proof.}}</ref> i.e., just a point. However, we expect the "correct" cohomology ring of <math>B\mathbb{G}_m</math> to be more like that of <math>\mathbb{C} P^\infty</math> as the ring should classify line bundles. Thus, the cohomology of <math>B\mathbb{G}_m</math> should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.
== Notes == {{reflist}}
== References == *{{cite journal |last=Behrend |first=K. |author-link=Kai Behrend |url=http://www.math.ubc.ca/~behrend/ladic.pdf |title=Derived l-adic categories for algebraic stacks |journal=Memoirs of the American Mathematical Society |volume=163 |year=2003|doi=10.1090/memo/0774 }}
Category:Algebraic geometry
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