Remarks on some Homological Problems regarding Infinite Integral Extensions
Abstract
Let R be an excellent local domain. R is said to be NBIM if ToriR(R+, k) = 0 for some i≥ d:=dim(R). Bhatt, Iyengar, and Ma ask if equi-characteristic zero NBIM rings are regular. If R is of positive characteristic, Asgharzadeh and Mahdavi conjecture that ExtiR(k,R∞) = 0 for some i>d implies that R is regular. It is an open question whether R+ and R∞ are m-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric NBIM rings are regular and solves the conjecture for F-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is R→ S finite and flat on the punctured spectrum and S is regular, this uses Cohen-Macaulayness of S+.
Related
BibTeX
@misc{asgharzadeh2026remarks,
title = {{Remarks on some Homological Problems regarding Infinite Integral Extensions}},
author = {Mohsen Asgharzadeh and Shravan Patankar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.09473},
eprint = {2608.09473},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.09473},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Remarks on some Homological Problems regarding Infinite Integral Extensions
AU - Mohsen Asgharzadeh
AU - Shravan Patankar
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.09473
UR - https://arxiv.org/abs/2608.09473
AB - Let R be an excellent local domain. R is said to be NBIM if Tor_i^R(R^+, k) = 0 for some i≥ d:=dim(R). Bhatt, Iyengar, and Ma ask if equi-characteristic zero NBIM rings are regular. If R is of positive characteristic, Asgharzadeh and Mahdavi conjecture that Ext^i_R(k,R^∞) = 0 for some i>d implies that R is regular. It is an open question whether R^+ and R^∞ are m-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric NBIM rings are regular and solves the conjecture for F-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is R→ S finite and flat on the punctured spectrum and S is regular, this uses Cohen-Macaulayness of S^+.
ER -