Characterizations of regular local rings in positive characteristics

Li, Jinjia

doi:10.1090/S0002-9939-07-09158-7

Characterizations of regular local rings in positive characteristics
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Proc. Amer. Math. Soc. 136 (2008), 1553-1558 Request permission

Abstract:

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.

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