Characterizations of regular local rings in positive characteristics

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher $\mathrm{Tor}$ counterparts ${\mathfrak{i}} t_i={\lim}_{n \to \infty} \ell(\mathrm{Tor}_i(k,{}^{f^n} R))/p^{nd}$. 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.