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 $\backslash mathbb\{Q\}\_l$.

To understand the problem that motivates the notion, consider the classifying stack $B\backslash mathbb\{G\}\_m$ over $\backslash operatorname\{Spec\}\; \backslash mathbf\{F\}\_q$. Then $B\backslash mathbb\{G\}\_m\; =\; \backslash operatorname\{Spec\}\; \backslash mathbf\{F\}\_q$ in the étale topology;{{harvnb|Behrend|2003|loc=Proposition 5.2.9; in particular, the proof.}} i.e., just a point. However, we expect the "correct" cohomology ring of $B\backslash mathbb\{G\}\_m$ to be more like that of $\backslash mathbb\{C\}\; P^\backslash infty$ as the ring should classify line bundles. Thus, the cohomology of $B\backslash mathbb\{G\}\_m$ should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.