论文标题
关于较高维基的Bruhat-tits理论
On Bruhat-Tits theory over a higher dimensional base
论文作者
论文摘要
让$ k $是一个完美的领域。假设$ k $的特征满足某些驯服假设\ eqref {tameness}。令$ \ Mathcal o _ {_ n}：= k \ llbracket z _ {_ 1}，\ ldots，z _ {_ n} \ rrbracket $，并设置$ k _ {_ _ n}：= = = \ text {fext {ftext {fract}〜\ co _ n}让$ g $是一种几乎简单的，简单地连接的Aggine Chevalley Group计划，具有最大的圆环$ t $和Borel子组$ b $。给定$ n $ -tuple $ {\ bf f} =（f _ {_ 1}，\ ldots，f _ {_ n}）$ g $ root Systems of $ g $上的凹入功能，如bruhat-titts \ cite \ cite \ cite \ cite {bruhattits1}} n} - 结合子组$ {\ tt p} _ {_ {_ {\ bf f}} \ subset g（k _ {_ n}）$} $ n = 1 $。我们表明，这些组是{\ IT示意图}，即它们是平滑的值{\ em quasi-affine}（resp。 $ z_i = 0 $由$ f_i $（分别由公寓点给出的凹函数总和）给出。这提供了具有自然专业特性的Bruhat-tits组方案的较高维度类似物。 In §\ref{mixedstuff}, under suitable assumptions on $k$ §\ref{charassum}, we extend all these results for a $n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the root system of $G$ replacing $\mathcal o _ {_ n} $ by $ {\ co} \ llbracket x _ {_ 1}，\ cdots，x _ {_ n} \ rrbracket $，其中$ \ co $是一个完整的离散估值环，带有特征性的特征性$ k $ p $ p $。在本文的最后一部分中，我们以char Zero的应用来构建某些天然群体方案，以构建一组奇妙的嵌入以及某些{\ tt 2-parahoric}组方案的家族，以最小化的表面奇异性分辨率，这些分辨率在\ cite {balaproc}中产生。
Let $k$ be a perfect field. Assume that the characteristic of $k$ satisfies certain tameness assumptions \eqref{tameness}. Let $\mathcal O_{_n} := k\llbracket z_{_1}, \ldots, z_{_n}\rrbracket$ and set $K_{_n} := \text{Fract}~\cO_{_n}$. Let $G$ be an almost-simple, simply-connected affine Chevalley group scheme with a maximal torus $T$ and a Borel subgroup $B$. Given a $n$-tuple ${\bf f} = (f_{_1}, \ldots, f_{_n})$ of concave functions on the root system of $G$ as in Bruhat-Tits \cite{bruhattits1}, \cite{bruhattits}, we define {\it {\tt n}-bounded subgroups ${\tt P}_{_{\bf f}}\subset G(K_{_n})$} as a direct generalization of Bruhat-Tits groups for the case $n=1$. We show that these groups are {\it schematic}, i.e. they are valued points of smooth {\em quasi-affine} (resp. {\em affine}) group schemes with connected fibres and {\it adapted to the divisor with normal crossing $z_1 \cdots z_n =0$} in the sense that the restriction to the generic point of the divisor $z_i=0$ is given by $f_i$ (resp. sums of concave functions given by points of the apartment). This provides a higher-dimensional analogue of the Bruhat-Tits group schemes with natural specialization properties. In §\ref{mixedstuff}, under suitable assumptions on $k$ §\ref{charassum}, we extend all these results for a $n+1$-tuple ${\bf f} = (f_{_0}, \ldots, f_{_n})$ of concave functions on the root system of $G$ replacing $\mathcal O_{_n}$ by ${\cO} \llbracket x_{_1},\cdots,x_{_n} \rrbracket$ where $\cO$ is a complete discrete valuation ring with a perfect residue field $k$ of characteristic $p$. In the last part of the paper, we give applications in char zero to constructing certain natural group schemes on wonderful embeddings of groups and also certain families of {\tt 2-parahoric} group schemes on minimal resolutions of surface singularities that arose in \cite{balaproc}.
