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;^[1] 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.