In algebraic geometry, the smooth topology is a certain

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...

, 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 t ...

. 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 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