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 Q l .
To understand the problem that motivates the notion, consider the classifying stack B G m over Spec F q . Then B G m = Spec F q in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of B G m to be more like that of C P ∞ as the ring should classify line bundles. Thus, the cohomology of B G m should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.
