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 ${\displaystyle \mathbb {Q} _{l}}$.

To understand the problem that motivates the notion, consider the classifying stack ${\displaystyle B\mathbb {G} _{m}}$ over ${\displaystyle \operatorname {Spec} \mathbf {F} _{q}}$. Then ${\displaystyle B\mathbb {G} _{m}=\operatorname {Spec} \mathbf {F} _{q}}$ in the étale topology;^[1] i.e., just a point. However, we expect the "correct" cohomology ring of ${\displaystyle B\mathbb {G} _{m}}$ to be more like that of ${\displaystyle \mathbb {C} P^{\infty }}$ as the ring should classify line bundles. Thus, the cohomology of ${\displaystyle B\mathbb {G} _{m}}$ should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.