关于亚纯下降的一个定理与原基本群的特殊化
A theorem on meromorphic descent and the specialization of the pro-étale fundamental group
摘要：
给出了${\rm Spf}(R)$上的Noether形式概形$\hat X$，其中$R$是完备的DVR，首先证明了沿$\hat{X}$的可能无限覆盖的亚纯下降定理.利用这一点，我们构造了一个特殊化函子，它是从特殊纤维的前向基本群的连续表示范畴到一般纤维上的$F$ -分裂层范畴的。这个特殊化函子部分地恢复了经典基本群的特殊化函子。我们还将域上有限型连通概型$X$的预基本群表示为自由群的余积和余积，以及$X$的不可约分支及其奇异轨迹的不可约分支的正规化的预基本群.
英文摘要：
Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the pro-étale fundamental group of the special fiber to the category of $F$-divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the pro-étale fundamental group of a connected scheme $X$ of finite type over a field as coproducts and quotients of the free group and the étale fundamental groups of the normalizations of the irreducible components of $X$ and those of its singular loci.
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