Remarks on some Homological Problems regarding Infinite Integral Extensions

Abstract

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $To{r}_i^R(R^{+},k)=0$ for some $i\ge d:=\dim (R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ex{t}_R^i(k,R^{\infty })=0$ 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 $R\to S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.