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.