2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/141589In 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 $ıt_i=\underset{n \to \infty}{\lim} ł(\tor_i(k,{}^{f^n} R))/p^{nd}$. 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.6 pagesCommutative Algebra13A35, 13D07, 13D25, 13H05text