Asymptotic vanishing conditions which force regularity in local rings of prime characteristic

TL;DR

This paper establishes that certain asymptotic vanishing conditions in prime characteristic local rings imply regularity, extending known results from low indices to higher ones using novel methods.

Contribution

It proves that vanishing of specific asymptotic invariants or Tor modules at higher indices guarantees regularity, generalizing prior results limited to low indices.

Findings

Vanishing of $ extgothic t_i(R)$ for some $i>0$ implies regularity.

Vanishing of $ ext{Tor}_i^R(k,R^+)$ for some $i>0$ implies regularity.

Results extend previous known cases for $i=1,2$ to higher indices.

Abstract

Let $(R,\m,k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_\dt$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i(R) = \lim_{e\to \8} {\length(H_i(F^e(G_\dt)))}/{p^{ed}}$. We show that if $\gothic t_i(R) = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\Tor_i^R(k,R^+) = 0$ for some $i>0$, then $R$ is regular (where $R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i = 1$ or 2 via completely different methods.