On generalized Hilbert-Kunz function and multiplicity

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 $R$. We exploit this fact to give a quick proof that if $R$ is a complete intersection of dimension $3$, then the Picard group of the punctured spectrum of $R$ is torsion-free. 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.