Séries de Poincaré et modules inertes. (Poincaré series and inert modules)
type: Item
instance of: scholarly article
title: Séries de Poincaré et modules inertes. (Poincaré series and inert modules)
zbMATH Open document ID: 0716.13011
DOI: 10.1016/0021-8693(90)90250-R
author: Jack Lescot
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let f: (S,\({\mathfrak n})\to (R,{\mathfrak m})\) be a homomorphism of noetherian local rings, which induces the identity map on their common residue field k, and let M be an R-module. The paper investigates the relation of the Poincaré series of M over R, \(P^ M_ R(t)=\sum_{i\geq 0}\dim_ kTor^ R_ i(M,k)t^ i \), and a similar formal power series over S. Of course, in order for \(P^ M_ R(t)\) to be defined, one needs the dimensions of the corresponding Tor's to be finite, and the author proves that this condition is equivalent to the finiteness of \(\dim_ kExt^ i_ R(k,M)\) for \(i\in {\mathbb{Z}}\), and to the finiteness of \(\dim_ kH(K\otimes_ RM)\), when K is the Koszul complex over a minimal system of generators of \({\mathfrak m}\). He also shows that if \(P^ M_ S(t)\) is defined, then so is \(P^ M_ R(t)\), and that the converse holds when S/\({\mathfrak m}S\) is artinian. These results partly depend on a rather general change of rings spectral sequence constructed in the paper, which in particular yields a coefficientwise inequality: \(P^ k_ R(t)\cdot P^ M_ S(t)\gg P^ k_ S(t)\cdot P^ M_ R(t).\) The author says an R-module M is inert by f if the inequality above becomes an equality. The study of inert modules, which is the main object of the paper, is linked to earlier work on the homology of local rings. Thus, it is shown that the homomorphism f is large [that is, the induced map \(f_*: Tor^ S(k,k)\to Tor^ R(k,k)\) is surjective, cf. \textit{G. Levin}, Math. Scand. 46, 209-215 (1980; Zbl 0472.13010)] if and only if every finite type R-module M is inert by f. On the other hand, if f is small [that is, \(f_*\) is injective, cf. the reviewer, J. Algebra 50, 400-453 (1978; Zbl 0395.13005)] then M is inert by f if and only if the induced map \(Tor^ S(M,k)\to Tor^ R(M,k)\) is injective. A particularly nice application of the ideas and techniques of the paper is given in the proof of the following result: If the ring R is Golod, and the module M is finitely generated, …
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13D02
zbMATH DE Number: 4179508
zbMATH Keywords: Golod rings
zbMATH Keywords: Poincaré series
zbMATH Keywords: Koszul complex
zbMATH Keywords: inert modules
zbMATH Keywords: homology of local rings
zbMATH Keywords: hypersurface ring
reviewed by: Luchezar L. Avramov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4179508