等特征零点绝对整闭包的Tor的消失
Vanishing of Tors of absolute integral closures in equicharacteristic zero
摘要：
证明了环$R$是正则的，如果对某些$i\geq 1$有$Tor_{i}^{R}(R^{+},k) = 0$，进一步假设$R$是在等特征零场$k$上生成的维数为$2$的$\mathbb{N}$ -分次环.这回答了Bhatt、Iyengar和Ma的一个问题。我们利用$R^{+}$上几乎所有的数学方法来推导Noether环$R$和有理曲面奇点的性质。此外，我们表明，$R^{+}$在等特征零是$m$ -adically理想（明智的）分离的，一个条件，出现在证明局部判据平坦。在维度$2$中，它是Ohm-Rush和相交平面。作为一个应用程序，我们表明，该假设可以是令人惊讶的空虚的$i \ll dim(R)$。我们表明，一个积极的回答一个老问题的Aberbach和Hochster也回答了这个问题。我们使用我们的技术，使一些评论安德烈和Fiorot关于'fpqc analgoues'的碎片的问题。
英文摘要：
We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a question of Bhatt, Iyengar, and Ma. We use almost mathematics over $R^{+}$ to deduce properties of the noetherian ring $R$ and rational surface singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is $m$-adically ideal(wise) separated, a condition which appears in the proof of local criterion for flatness. In dimension $2$ it is Ohm-Rush and intersection flat. As an application we show that the hypothesis can be astonishingly vacuous for $i \ll dim(R)$. We show that a positive answer to an old question of Aberbach and Hochster also answers this question. We use our techniques to make some remarks on a question of André and Fiorot regarding `fpqc analgoues' of splinters.
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