On generalized Hilbert-Kunz function and multiplicity

Abstract

Let \((R,\mathfrak m)\) be a local ring of characteristic \(p>0\) and \(M\) a finitely generated \(R\)-module. In this note we consider the limit: \(\lim_{n\to \infty} \frac{\ell(H^0_{\mathfrak m}(F^n(M)))}{p^{n\dim R}} \) where \(F(-)\) is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when \(R\) is excellent and has isolated singularity. Furthermore, if \(R\) is a complete intersection, then the limit is 0 if and only if the projective dimension of \(M\) is less than the Krull dimension of \(R\). Our results work quite generally for other homological functors and can be used to prove that certain limits recently studied by Brenner exist over projective varieties.