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 $B{𝔾}_m$ over $\mathrm{Spec}{𝐅}_q$. Then $B{𝔾}_m=\mathrm{Spec}{𝐅}_q$ 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 $ℂP^{\mathrm{\infty }}$ 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.

References

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