Finite torsors over strongly $F$-regular singularities
Finite torsors over strongly $F$-regular singularities
复制标题
强 $F$ 正则奇点上的有限扭转
DOI：
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发表时间：
2017
期刊：
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通讯作者：
Javier Carvajal
中科院分区：
文献类型：
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作者：
Javier Carvajal
我们研究了强谱的大开数上的有限扭量
$F$-正常的细菌，不会延伸到整个光谱的扭体。让我们
$(R，mathfrak{m}，k)$是强$F$-正则$k$-芽，其中$k$是
特征为$p>；0$的代数闭域。我们证明了一个
有限局部覆盖$R子集R^{STAR}$使得$R^{STAR}$是一个强
$F$-正则$k$-芽和：对所有具有可解的有限代数群$G/k$
中性组件，在$mathm{Spec}的大开口上的每个$G$-扭力
R^{star}$处处扩张到$G$-扭子。为了实现这一点，我们获得了一个
有限局部下$F$-签名的广义变换规则
分机。这个公式被用来证明$mathm{CL}的挠率
R$以$1/S(R)$为界。通过取圆锥体，我们得出结论，Picard群
在全球范围内，$F$-常规品种是无扭矩的。同样，它表明
$mathbb{q}$-Gorenstein强$F$-正则奇点的正则覆盖
是很强的$F$-规则。
We investigate finite torsors over big opens of spectra of strongly
$F$-regular germs that do not extend to torsors over the whole spectrum. Let
$(R,mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an
algebraically closed field of characteristic $p>0$. We prove the existence of a
finite local cover $R subset R^{star}$ so that $R^{star}$ is a strongly
$F$-regular $k$-germ and: for all finite algebraic groups $G/k$ with solvable
neutral component, every $G$-torsor over a big open of $mathrm{Spec}
R^{star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain a
generalized transformation rule for the $F$-signature under finite local
extensions. Such formula is used to show that that the torsion of $mathrm{Cl}
R$ is bounded by $1/s(R)$. By taking cones, we conclude that the Picard group
of globally $F$-regular varieties is torsion-free. Likewise, it shows that
canonical covers of $mathbb{Q}$-Gorenstein strongly $F$-regular singularities
are strongly $F$-regular.