Vanishing of Tors of absolute integral closures in equicharacteristic zero

2022/12/18 by Patankar, Shravan · 2 citations
#13A35 #13D22 #14B05 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.09025

Abstract

We show that a ring R is regular if Tor_i^R(R⁺,k) = 0 for some i≥ 1 assuming further that R is a 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 ≪ 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.

Cited by

• Non-Noetherian Bass and Betti numbers
• A study of perfectoid rings via Galois cohomology

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