On a conjecture of Murthy
type: Item
instance of: scholarly article
title: On a conjecture of Murthy
zbMATH Open document ID: 1412.13016
author: Mrinal Kanti Das
published in: Advances in Mathematics
publication date: 2018-06-26T00:00:00Z
review text: An old question of \textit{M. P. Murthy} asks whether and ideal $I\subset k[x_1,\dots, x_r]=R$ can be generated by the same number of elements required to generate $I/I^2$ [in: Conf. commut. Algebra, Kingston196--211 (1975; Zbl 0354.14015)]. If we denote by $\mu(M)$, the minimal number of generators of a module $M$, the question asks whether $\mu(I)=\mu(I/I^2)$. The reviewer had given an affirmative answer when $\mu(I/I^2)\geq\dim R/I +2$ [\textit{N. Mohan Kumar}, Invent. Math. 46, 225--236 (1978; Zbl 0395.13009)]. Recently a complete affirmative answer was announced by \textit{J. Fasel} [Ann. Math. (2) 184, No. 1, 315--331 (2016; Zbl 1372.13014)], but the proof had gaps and thus remains incomplete; and the question remained open in general. In the paper under review, the author improves the earlier result of the reviewer in the case where $k$ is the algebraic closure of a finite field and with the weaker hypothesis, $\mu(I/I^2)\geq\dim R/I+1$.
reviewed by: N. Mohan Kumar
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 19A15
Mathematics Subject Classification ID: 14R99
zbMATH DE Number: 6894537
zbMATH Keywords: polynomial rings
zbMATH Keywords: number of generators of an ideal
zbMATH Keywords: number of generators of conormal module
Wikidata QID: Q123302401
MaRDI profile type: Publication
full work available at URL: j.aim.2018.04.012
OpenAlex ID: W2800209643
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DOI: 10.1016/J.AIM.2018.04.012
description: scientific article; zbMATH DE number 6894537