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; 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.

References

• Behrend, K. (2003). "Derived l-adic categories for algebraic stacks" (PDF). Memoirs of the American Mathematical Society. 163.
• Laumon, Gérard; Moret-Bailly, Laurent (2000), Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 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).