  In this note, we provide several characterizations of regular local rings in
positive characteristics, in terms of the Hilbert-Kunz multiplicity and its
higher $\tor$ counterparts $\i t_i=\underset{n \to \infty}{\lim}
\l(\tor_i(k,{}^{f^n} R))/p^{nd}$. We also apply the characterizations to
improve a recent result by Bridgeland and Iyengar in the characteristic $p$
case. Our proof avoids using the existence of big Cohen-Macaulay modules, which
is the major tool in the proof of Bridgeland and Iyengar.
