The arc-topology

By B. Bhatt, A. Mathew. 2018

71 citations recorded by the source corpus.

We study a Grothendieck topology on schemes which we call the $\mathrm{arc}$-topology. This topology is a refinement of the $v$-topology (the pro-version of Voevodsky's $h$-topology) where covers are tested via rank $\leq 1$ valuation rings. Functors which are $\mathrm{arc}$-sheaves are forced to satisfy a variety of glueing conditions such as excision in the sense of algebraic $K$-theory. We show that \'etale cohomology is an $\mathrm{arc}$-sheaf and deduce various pullback squares in \'etale cohomology. Using $\mathrm{arc}$-descent, we reprove the Gabber-Huber affine analog of proper base change (in a large class of examples), as well as the Fujiwara-Gabber base change theorem on the \'etale cohomology of the complement of a henselian pair. As a final application we prove a rigid analytic version of the Artin-Grothendieck vanishing theorem from SGA4, extending results of Hansen.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/459261e0b279ce9d85e1963c9bfb329da2d7b340

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Prisms and prismatic cohomology (2019, methodology): “etermined by dX∈Ω1 T/R. 8.2. arc-descent. In this section, we use Proposition 8.5 to deduce some very strong descent results for the association S→Sperfd, formulated in terms of the arc-topology from [BM18]4. 4We are trying to achieve optimal results here and thus use the arc-topology. Getting simil”