完全可分闭概形上的几何
Geometry on totally separably closed schemes
AI总结：
本文证明基域上拟紧分离概形关于Nisnevich拓扑的Cech上同调与层上同调一致，推广了Artin结果，关键证明分离代数概形绝对整闭包上不可约闭子集交集的不可约性。
摘要：
我们证明，对于基域上的拟紧分离概形，关于Nisnevich拓扑的Cech上同调与层上同调一致。这是Artin结果的局部推广，该结果指出，对于诺特概形，在假设每个有限子集具有仿射开邻域（AF性质）的条件下，关于平展拓扑的上述等式成立。我们的关键结果是，在分离代数概形的绝对整闭包上，任意两个不可约闭子集的交集仍不可约。我们通过建立适用于概形逆向极限的一般性修改与收缩结果来证明这一点。在此过程中，我们刻画了关于各种Grothendieck拓扑无环的概形，研究了所有局部环均为严格亨塞尔环的概形，并分析了严格局部化的纤维积。
英文摘要：
We prove, for quasicompact separated schemes over ground fields, that Cech cohomology coincides with sheaf cohomology with respect to the Nisnevich topology. This is a partial generalization of Artin's result that for noetherian schemes such an equality holds with respect to the etale topology under the assumption that every finite subset admits an affine open neighborhood (AF-property). Our key result is that on the absolute integral closure of separated algebraic schemes, the intersection of any two irreducible closed subsets remains irreducible. We prove this by establishing general modification and contraction results adapted to inverse limits of schemes. Along the way, we characterize schemes that are acyclic with respect to various Grothendieck topologies, study schemes all whose local rings are strictly henselian, and analyze fiber products of strict localizations.
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