Mathematical Research Letters

Volume 15 (2008)

Number 4

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

Pages: 815 – 820



Ian M. Aberbach (University of Missouri)

Jinjia Li (Syracuse University)


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.

Published 1 January 2008