A Frobenius characterization of finite projective dimension over complete intersections
type: Item
instance of: scholarly article
title: A Frobenius characterization of finite projective dimension over complete intersections
zbMATH Open document ID: 0977.13005
DOI: 10.1007/PL00004783
published in: Mathematische Zeitschrift
publication date: 2000-07-18T00:00:00Z
review text: Let \(R\) be a Noetherian local ring of dimension \(d\) which is a complete intersection and contains a field of characteristic \(p\). Let \(F(-)\) denote the Frobenius functor and \(\ell\) the length function. Let \(M\) be an \(R\)-module of finite length. In this paper the author shows that \(M\) has finite projective dimension if and only if \(\ell(F^n(M))=\ell(M)p^{nd}\) for all \(n\geq 0\). This may be viewed as the converse to a theorem of \textit{S. Dutta} [J. Algebra 85, 424-448 (1983; Zbl 0527.13014)]. As a corollary the author gets that, in a complete intersection \((R,\mathfrak m)\), an \(\mathfrak m\)-primary ideal \(I\) has finite projective dimension if and only if its Hilbert-Kunz multiplicity equals the length of \(R/I\).
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13D40
zbMATH DE Number: 1406451
zbMATH Keywords: Frobenius functor
zbMATH Keywords: projective dimension
zbMATH Keywords: Hilbert-Kunz multiplicity
zbMATH Keywords: characteristic \(p\)
author: Claudia Miller
MaRDI profile type: Publication
full work available at URL: pl00004783
OpenAlex ID: W2065599615
description: scientific article; zbMATH DE number 1406451