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 l. To understand the problem that motivates the notion, consider the classifying stack B𝔾m over SpecFq. Then B𝔾m=SpecFq in the étale topology;[1] i.e., just a point. However, we expect the "correct" cohomology ring of B𝔾m to be more like that of P as the ring should classify line bundles. Thus, the cohomology of B𝔾m 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

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