  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.
