Vanishing of Tors of absolute integral closures in equicharacteristic zero

Abstract

We show that R R is regular if T o r i R ( R + , k ) = 0 Tor_{i}^{R}(R^{+},k) = 0 for some i ≥ 1 i\geq 1 assuming further that R R is a N \mathbb {N} -graded ring of dimension 2 2 finitely generated over an algebraically closed equicharacteristic zero field k k . This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R + R^{+} to deduce properties of the noetherian ring R R and rational surface singularities. Moreover we observe that R + R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is m m -adically ideal(wise) separated, (weakly) intersection flat, and Ohm-Rush. As an application we show that the hypothesis can be astonishingly vacuous for i ≪ d i m ( R ) i \ll dim(R) . We show that a positive answer to an old question of Aberbach and Hochster [J. Pure Appl. Algebra 122 (1997), pp. 171–184] also answers this question and we use our techniques to study a question of André and Fiorot [Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), pp. 81–114] regarding ‘fpqc analogues’ of splinters.