弧拓扑
The arc-topology
摘要：
我们研究了一个Grothendieck拓扑的计划，我们称之为$\mathrm{arc}$ -拓扑。该拓扑是$v$ -拓扑（Voevodsky的$h$ -拓扑的前版本）的改进，其中覆盖经由秩$\leq 1$赋值环来测试。作为$\mathrm{arc}$ -层的函子必须满足各种粘合条件，例如代数$K$ -理论意义上的切除。我们证明了étale上同调是一个$\mathrm{arc}$ -层，并推导了平展上同调中的各种拉回方.利用$\mathrm{arc}$ -下降，我们重新证明了正常基变换的Gabber-Huber仿射模拟（在一个大类的例子中），以及关于Henselian对的补的平展上同调的Fujiwara-Gabber基变换定理.作为最后的应用程序，我们证明了严格的解析版本的Artin-Grothendieck消灭定理从SGA 4，扩展汉森的结果。
英文摘要：
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 étale cohomology is an $\mathrm{arc}$-sheaf and deduce various pullback squares in étale 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 étale 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.
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