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.

Notes

1. Behrend, Proposition 5.2.9; in particular, the proof.

References

• Behrend, K. Derived l-adic categories for algebraic stacks. Memoirs of the American Mathematical Society Vol. 163, 2003.
• Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 39, Berlin, New York: Springer-Verlag, ISBN 978-3-540-65761-3, MR 1771927 Unfortunately this book uses the incorrect assertion that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. Some of these errors were fixed by Olsson (2007).