In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than

étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale ...

. Its main use is to define the cohomology of an

algebraic stack In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's repr ...

with coefficients in, say, the étale sheaf

\mathbb_l

. To understand the problem that motivates the notion, consider the

classifying stack In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. Th ...

B\mathbb_m

over

\operatorname \mathbf_q

. Then

B\mathbb_m = \operatorname \mathbf_q

in the étale topology; i.e., just a point. However, we expect the "correct" cohomology ring of

B\mathbb_m

to be more like that of

\mathbb P^\infty

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

B\mathbb_m

should be defined using smooth topology for formulae like

Behrend's fixed point formula In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz trace formula to a smooth algebraic stack over a finite field conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical one, the ...

to hold.

Notes

References

* * 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 . Algebraic geometry {{algebraic-geometry-stub