Smooth topology

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 $$\mathbb{Q}_l$$.

To understand the problem that motivates the notion, consider the classifying stack $$B\mathbb{G}_m$$ over $$\operatorname{Spec} \mathbf{F}_q$$. Then $$B\mathbb{G}_m = \operatorname{Spec} \mathbf{F}_q$$ in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of $$B\mathbb{G}_m$$ to be more like that of $$\mathbb{C} P^\infty$$ as the ring should classify line bundles. Thus, the cohomology of $$B\mathbb{G}_m$$ should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.