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Smooth topology

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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

  1. Behrend 2003, Proposition 5.2.9; in particular, the proof.

References


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