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 [math]\displaystyle{ \mathbb{Q}_l }[/math]. To understand the problem that motivates the notion, consider the classifying stack [math]\displaystyle{ B\mathbb{G}_m }[/math] over [math]\displaystyle{ \operatorname{Spec} \mathbf{F}_q }[/math]. Then [math]\displaystyle{ B\mathbb{G}_m = \operatorname{Spec} \mathbf{F}_q }[/math] in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of [math]\displaystyle{ B\mathbb{G}_m }[/math] to be more like that of [math]\displaystyle{ \mathbb{C} P^\infty }[/math] as the ring should classify line bundles. Thus, the cohomology of [math]\displaystyle{ B\mathbb{G}_m }[/math] 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". Memoirs of the American Mathematical Society 163. http://www.math.ubc.ca/~behrend/ladic.pdf. 
• 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  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).