Vanishing of Tors of absolute integral closures in equicharacteristic zero

TL;DR

This paper proves that in equicharacteristic zero, the vanishing of certain Tor modules over the absolute integral closure characterizes regularity of the ring, using almost mathematics and properties of rational surface singularities.

Contribution

It establishes a new criterion for regularity based on Tor vanishing over the absolute integral closure in equicharacteristic zero, answering a question by Bhatt, Iyengar, and Ma.

Findings

Vanishing of Tor implies regularity in specific graded rings.

Absolute integral closure is m-adically separated and flat in dimension 2.

Results apply to rational surface singularities and relate to old conjectures.

Abstract

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 Andr\'e and Fiorot regarding `fpqc analgoues' of splinters.