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 ${\mathbb{\mathbb{Q}}}_l$. To understand the problem that motivates the notion, consider the classifying stack $B{\mathbb{𝔾}}_m$ over $\mathrm{Spec}{\mathbf{𝐅}}_q$. Then $B{\mathbb{𝔾}}_m=\mathrm{Spec}{\mathbf{𝐅}}_q$ 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 $\mathbb{ℂ}{P}^{\mathrm{\infty }}$ 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". 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).