论文标题
扩展Torsors和Prime至$ P $基本集团计划
Extension of torsors and prime to $p$ fundamental group scheme
论文作者
论文摘要
让$ r $是分数字段$ k $的离散评估戒指。令$ x $为适当且忠实的平面$ r $ -scheme，并在x（r）$中的$ x \ extrough，带有连接和减少的通用光纤$x_η$。令$ f：y \ rightarrowx_η$为有限的nori-dred $ g $ -torsor。在本文中，我们提供了一个有用的标准，可以将$ f：y \ rightarrowx_η$扩展到$ x $以上的torsor。此外，在特定情况下，$ r $是一个完整的离散估值环$ p> 0 $和$ x \ to \ to \ text {spec}（r）$很平稳，我们应用我们的标准来证明自然形态$ψ^{（p'（p'p'（p'）}} π（x，x）_η^{（p'）} $ $x_η$的prime-to-p $基本组方案与$ x $ prime-to-$ p $ P $基本组方案的通用纤维是同构的。这概括了埃塔尔基本小组的众所周知的结果。所使用的方法纯粹是坦纳基安。
Let $R$ be a discrete valuation ring with fraction field $K$. Let $X$ be a proper and faithfully flat $R$-scheme, endowed with a section $x \in X(R)$, with connected and reduced generic fibre $X_η$. Let $f: Y \rightarrow X_η$ be a finite Nori-reduced $G$-torsor. In this paper we provide a useful criterion to extend $f: Y \rightarrow X_η$ to a torsor over $X$. Furthermore in the particular situation where $R$ is a complete discrete valuation ring of residue characteristic $p>0$ and $X\to \text{Spec}(R)$ is smooth we apply our criterion to prove that the natural morphism $ψ^{(p')}: π(X_η,x_η)^{(p')}\to π(X,x)_η^{(p')}$ between the prime-to-$p$ fundamental group scheme of $X_η$ and the generic fibre of the prime-to-$p$ fundamental group scheme of $X$ is an isomorphism. This generalizes a well known result for the étale fundamental group. The methods used are purely tannakian.
