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: ${\lim }_{n\to \infty }\frac{\ell (H_{\mathfrak{m}}^0(F^n(M)))}{p^{n\dim R}}$ 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.

References in corpus (3)

• The Frobenius endomorphism and homological dimensions
• Irrational Hilbert-Kunz multiplicities
• Multiplicities of graded families of linear series and ideals

Cited by in corpus (10)

• Finite F-type and F-abundant modules
• Generalized Hilbert-Kunz function in graded dimension two
• Hilbert-Kunz functions of surface rings of type ADE
• On the existence of $F$-thresholds and related limits
• Local Okounkov bodies and limits in prime characteristic
• Some computations of generalized Hilbert-Kunz function and multiplicity
• $F$-Invariants of Stanley-Reisner Rings
• Length of local cohomology of powers of ideals
• On the (LC) conjecture
• A note on Hilbert-Kunz multiplicity