Local rings over which all modules have rational Poincaré series
type: Item
instance of: scholarly article
title: Local rings over which all modules have rational Poincaré series
zbMATH Open document ID: 0794.13010
DOI: 10.1016/0022-4049(94)90134-1
author: Luchezar L. Avramov
published in: Journal of Pure and Applied Algebra
publication date: 1994-08-31T00:00:00Z
review text: Soient \(R\) un anneau commutatif local d'idéal maximal \(M\) et corps résiduel \(k\), \(E\) un \(R\)-module de type fini et \(P^ R_ E(n)=\sum^ \infty_{n=0} b^ R_ n (E)t^ n\) la série de Poincaré du \(R\)- module \(E\), où \(b^ R_ n(E)=\dim_ k \text{Tor}^ R_ n (E,k)\) sont les nombres de Betti. L'A. démontre les résultats principaux suivants: Si l'algèbre de Lie homotopique \(\Pi^*(R)\) contient une sous-algèbre de Lie libre graduée de codimension finie, il existe un polynôme \(\text{Den}^ R(t) \in \mathbb{Z} [t]\) tel que our tout \(R\)- module de type fini \(E\) la série \(\text{Den}^ R (t) P^ R_ E(t) \in \mathbb{Z} [t]\). Si, de plus, la sous-algèbre de Lie libre contient \(\Pi^ j(R)\) pour \(j>n\), alors \(\text{Den}^ R (t)P^ R_ k(t)\) divise \(\prod_{0 \leq 2i+1 \leq n} (1+t^{2i+1})^{e_{2i+1}}\), où \(e_ i=\dim_ k \pi^ i(R)\) est la deviation d'ordre \(i\) de \(R\). Si la fonction \(b^ R_ E\) est polynômiale pour tout \(R\)-module de type fini \(E\), alors \(R\) est un anneau d'intersection complète de codimension \(\leq q(R)+1\), où \(q(R)=(e(e+1)/2)-\dim_ k (M^ 2/M^ 3)\). Si \(R\) est une intersection complète de multiplicité minimale, alors \(b^ R_ E\) est une fonction polynômiale.
reviewed by: Nicolae Radu
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13C40
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 500564
zbMATH Keywords: complete intersection
zbMATH Keywords: Poincaré series
zbMATH Keywords: homotopic Lie algebra
zbMATH Keywords: Betti numbers
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90134-1
OpenAlex ID: W1963516280
cites work: A counterexample to a conjecture of Serre
cites work: Small homomorphisms of local rings
cites work: Obstructions to the Existence of Multiplicative Structures on Minimal Free Resolutions
cites work: Free Lie Subalgebras of the Cohomology of Local Rings
cites work: Q3344077
cites work: Q3717179
cites work: Modules of finite virtual projective dimension
cites work: Q3973354
cites work: Q5201919
cites work: Q3715232
cites work: Poincaré series of modules over local rings of small embedding codepth or small linking number
cites work: Q3903156
cites work: Q3701517
cites work: Generic free resolutions and a family of generically perfect ideals
cites work: Homological Algebra on a Complete Intersection, with an Application to Group Representations
cites work: Boundedness Versus Periodicity Over Commutative Local Rings
cites work: Q4117932
cites work: Q5733613
cites work: A Change of Ring Theorem with Applications to Poincaré Series and Intersection Multiplicity.
cites work: On the Hilbert series of the homology of differential gaded algebras.
cites work: On the deviations of a local ring.
cites work: Q5610428
cites work: The Poincaré series of a codimension four Gorenstein ring is rational
cites work: Poincaré-Betti Series are Primitive Recursive
cites work: ON THE DERIVED CATEGORY ANDK-FUNCTOR OF COHERENT SHEAVES ON INTERSECTIONS OF QUADRICS
cites work: The Poincaré series of every finitely generated module over a codimension four almost complete intersection is a rational function
cites work: Algèbres connexes et homologie des espaces de lacets
cites work: Finitely generated Ext algebras.
cites work: Q3726293
cites work: On the structure of Hopf algebras
cites work: Q3674791
cites work: Q1313788
cites work: Multiplicity andt-isomultiple ideals
cites work: Algebra structures on some canonical resolutions
cites work: Growth of Betti numbers of modules over local rings of small embedding codimension or small linkage number
cites work: Homology of Noetherian rings and local rings
description: scientific article; zbMATH DE number 500564