Generalized Hilbert-Kunz function in graded dimension two

Abstract

We prove that the generalized Hilbert-Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$-domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^2+\gamma (q)$, with rational generalized Hilbert-Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\gamma (q)$. Moreover we prove that if $R$ is a $\mathbb{Z}$-algebra, the limit for $p\to +\infty$ of the generalized Hilbert-Kunz multiplicity $e_{gHK}^{R_p}(M_p)$ over the fibers $R_p$ exists and it is a rational number.

Shortened the proofs of Lemma 1.1 and Lemma 1.3; improved Remark 1.4 and Example 3.6; improved exposition; updated references

References in corpus (5)

• Maximal Cohen-Macaulay modules over surface singularities
• On generalized Hilbert-Kunz function and multiplicity
• Hilbert-Kunz multiplicity and reduction mod p
• Some computations of generalized Hilbert-Kunz function and multiplicity
• On the (LC) conjecture

Cited by in corpus (2)

• Length of local cohomology of powers of ideals
• On the (LC) conjecture