On the Hilbert-Kunz function and Koszul homology
type: Item
instance of: scholarly article
title: On the Hilbert-Kunz function and Koszul homology
zbMATH Open document ID: 0829.13004
published in: Journal of Algebra
publication date: 1996-01-17T00:00:00Z
review text: Let \(M\) be a finitely generated module over a local noetherian ring \((R,m)\) of characteristic \(p > 0\), and let \(I\) be an \(m\)-primary ideal. Let \(q = p^n\) and \(I^{[q]} = (i^q : i \in I)\). The Hilbert-Kunz function of \(M\) is the function \(n \mapsto l_n (M) = l_R (M/I^{[q]}M)\). Some results of the paper: If \(R = kX,Y\), then \(l_n (M) = (\text{rk}M) p^{2n} + Cp^n + \Delta_n\), where \(C\) is an integer \(\geq 0\) and \(\Delta_n\) is an eventually periodic function of \(n\). If \(R = kX_1, \ldots, X_d\), \(d \geq 3\), then \(l_n (M) = (\text{rk} M)p^{dn} + Cp^{(d - 1)n} + O(p^{(d - 2)n})\), where \(C\) is a real number \(\geq 0\). Several open problems are stated.
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 794340
zbMATH Keywords: formal power series ring
zbMATH Keywords: characteristic \(p\)
zbMATH Keywords: finitely generated module
zbMATH Keywords: noetherian ring
zbMATH Keywords: Hilbert-Kunz function
author: Maria Contessa
MaRDI profile type: Publication
full work available at URL: jabr.1995.1212
OpenAlex ID: W1976633127
DOI: 10.1006/JABR.1995.1212
description: scientific article; zbMATH DE number 794340