Smooth topology explained

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} .B G _{m} over \operatorname{Spec} F _{q} . Then B G _{m} = \operatorname{Spec} F _{q} in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of B G _{m} to be more like that of C P ^{infty} as the ring should classify line bundles. Thus, the cohomology of B G _{m} should be defined using smooth topology for formulae like Behrend's fixed point formula to hold.

References

• Behrend . K. . Derived l-adic categories for algebraic stacks . Memoirs of the American Mathematical Society . 163 . 2003.