Remarks on some Homological Problems regarding Infinite Integral Extensions

Abstract

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-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 $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.

Disclosure

“hdavi, Shiji Lyu, and Sridhar Venkatesh for conversations. The main results of this paper were conceived after progress on Question (Q1 ) in [6] and we would like to thank the authors for their continued interest. The second author used GPT-5.6 Sol (by Open AI) to assist in finding references and for background proofs of routine facts from algebraic geometry, such as existence of endo- morphisms on toric varieties. No AI was used in proofs or discovery of the main results. All wr”

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2026-08-23T05:07:34.078320Z

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