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- Title
Vanishing of Tors of absolute integral closures in equicharacteristic zero.
- Authors
Patankar, Shravan
- Abstract
We show that 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 algebraically closed equicharacteristic zero field k. This answers a question of Bhatt, Iyengar, and Ma [Comm. Algebra 47 (2019), pp. 2367–2383]. We use almost mathematics over R^{+} to deduce properties of the noetherian ring R and rational surface singularities. Moreover we observe that R^{+} in equicharacteristic zero has a rich module-theoretic structure; it is 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 \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.
- Subjects
NOETHERIAN rings; ALGEBRA; INTEGRALS; MATHEMATICS
- Publication
Transactions of the American Mathematical Society, Series B, 2024, Vol 11, p98
- ISSN
2330-0000
- Publication type
Article
- DOI
10.1090/btran/174