An application of a theorem of E. Davis to Koszul complexes
type: Item
instance of: scholarly article
title: An application of a theorem of E. Davis to Koszul complexes
zbMATH Open document ID: 0652.13011
DOI: 10.1007/BF01196506
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \(x_ 1,...,x_ r\) be a multiplicity system on a Noetherian R-module M of dimension d, where R is a commutative ring, and let \(H_ iK(x_ 1,...,x_ r| M)\) be the corresponding i-th homology module of the Koszul complex \(K(x_ 1,...,x_ r| M)\). In an earlier note [J. Lond. Math. Soc., II. Ser. 34, 411-416 (1986; Zbl 0608.13005)], the author proved that the function \({\mathbb{N}}\to {\mathbb{N}}_ 0\) given by \(n\to \text{length}_ RH_ iK(x^ n_ 1,...,x^ n_ r| M)\) is bounded above by a polynomial of degree at most min(d,r-i). In this note a shorter proof is given using that theorem of Davis which states that when R is a local ring there exists \(a_ 2,...,a_ r\) in R such that \(x_ 1+\sum^{r}_{2}a_ rx_ r\) is in no non-maximal prime ideal of Ass(M).
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 4063237
zbMATH Keywords: homology module of the Koszul complex
author: J.-L. García Roig
reviewed by: David Kirby
MaRDI profile type: Publication
cites work: Q4723864
cites work: On Polynomial Bounds for the Koszul Homology of Certain Multiplicity Systems
cites work: Q5542855
full work available at URL: https://doi.org/10.1007/bf01196506
OpenAlex ID: W303771730
description: scientific article; zbMATH DE number 4063237