论文标题
消失的均等零中绝对积分封闭的tor
Vanishing of Tors of absolute integral closures in equicharacteristic zero
论文作者
论文摘要
我们表明，如果$ tor_ {i}^{r}（r^{+}，k）= 0 $，对于某些$ i \ geq 1 $，则环$ r $是常规的，假设$ r $是$ \ mathbb {n} $ - dimension $ 2 $ 2 $的$ 2 $ 2 $ agity Equi-characteristic $ k $ k $ k $ k k $ k $ k $ k $ k $ k $ k $ 2。这回答了Bhatt，Iyengar和MA的问题。我们在$ r^{+} $上使用几乎数学来推断Noetherian Ring $ r $的属性和理性的表面奇点。此外，我们表明，在qui-thacteristic零中，$ r^{+} $是$ m $ - 原理理想（明智），这种情况出现在局部平面标准的证明中。在尺寸$ 2 $中，它是欧姆式的，交点平坦。作为一个应用程序，我们表明，对于$ i \ ll dim（r）$，假设可能令人惊讶。我们表明，阿伯巴赫和霍克斯特的一个古老问题的积极答案也回答了这个问题。我们使用我们的技术就安德烈和菲奥罗特的“ fpqc肛门”碎片的问题发表了一些评论。
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.
