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

Q

l

{\displaystyle \mathbb {Q} _{l}}

.

To understand the problem that motivates the notion, consider the classifying stack

B

G

m

{\displaystyle B\mathbb {G} _{m}}

over

Spec

⁡

F

q

{\displaystyle \operatorname {Spec} \mathbf {F} _{q}}

. Then

B

G

m

=

Spec

⁡

F

q

{\displaystyle 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

G

m

{\displaystyle B\mathbb {G} _{m}}

to be more like that of

C

P

∞

{\displaystyle \mathbb {C} P^{\infty }}

as the ring should classify line bundles. Thus, the cohomology of

B

G

m

{\displaystyle B\mathbb {G} _{m}}

should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.

Notes

References

Behrend, K. (2003). "Derived l-adic categories for algebraic stacks" (PDF). Memoirs of the American Mathematical Society. 163.