A proof of Solomon's second conjecture on local zeta functions of orders
type: Item
instance of: scholarly article
title: A proof of Solomon's second conjecture on local zeta functions of orders
zbMATH Open document ID: 1025.16011
DOI: 10.1016/S0021-8693(02)00548-3
author: Osamu Iyama
published in: Journal of Algebra
publication date: 2003-04-02T00:00:00Z
review text: Let \(R\) be a complete discrete valuation domain with quotient field \(K\) and residue field \(k\) such that \(|k|=q<\infty\). Let \(\Lambda\) be an \(R\)-order in a semisimple \(K\)-algebra \(A\), and let \(V\) be a finite dimensional \(A\)-module. By the Jordan-Zassenhaus theorem, there is a finite representing system \(I\) of full \(\Lambda\)-lattices in \(V\). For \(L,M\in I\), consider the zeta function \[ Z_{LM}(s)=\sum(L:N)^{-s}, \] where \(N\) runs through the \(\Lambda\)-submodules of finite index in \(L\) which are isomorphic to \(M\). Solomon has shown that the \(|I|\times|I|\) matrix \(\mathbb{Z}_\Lambda(V;s)=(Z_{LM}(s))\) has an inverse with coefficients in \(\mathbb{Z}[q^{-s}]\). His second conjecture (1978) states that the determinant of \(\mathbb{Z}_\Lambda(V;s)\) is of the form \[ \prod(1-q^{a_i-b_is})^{-1} \] with \(a_i\in\mathbb{N}\) and \(b_i\in\mathbb{N}\setminus\{ 0\}\). The author proves this conjecture by an inductive method which already led to a solution of two other important problems in representation theory. Firstly, he obtained a characterization of the Auslander-Reiten quivers of representation-finite orders [\(\tau\)-categories, I, II, III, 72pp., Algebr. Represent. Theory (to appear)]. Secondly, he proved a conjecture of Ringel-Yamagata, which implies that the representation dimension of any Artinian algebra is finite [Proc. Am. Math. Soc. 131, No. 4, 1011-1014 (2003; Zbl 1018.16010)]. For the proof of Solomon's second conjecture, the author considers the full subcategory \(\mathcal C_V=\text{add }\mathcal X\) of \(\Lambda\text{-}\mathbf{lat}\), where \(\mathcal X\) consists of the finitely generated \(\Lambda\)-submodules of \(V\). By the Jordan-Zassenhaus theorem, there are only finitely many isomorphism-classes of indecomposable objects in \(\mathcal C_V\). He proves that there exists a chain \(\mathcal C_V=\mathcal C_m\supset\mathcal C_{m-1}\supset\cdots\supset\mathcal C_0=\Gamma\text{-}\mathbf{lat}\) of full subcategories with …
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 11S45
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 1891314
zbMATH Keywords: orders in semisimple algebras
zbMATH Keywords: indecomposable lattices
zbMATH Keywords: zeta functions
zbMATH Keywords: submodules of finite index
zbMATH Keywords: determinants
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: representation finite orders
zbMATH Keywords: hereditary orders
Wikidata QID: Q123165214
MaRDI profile type: Publication
cites work: Q3921357
cites work: Zeta functions of arithmetic orders and Solomon's conjectures
cites work: Q3949137
cites work: The prime ideal theorem in non-commutative arithmetic
cites work: New asymptotic formulas for the distribution of left ideals of orders.
cites work: Zeta functions and composition factors for arithmetic orders
cites work: Algebraic stratification in representation categories
cites work: Q3996909
cites work: Solomon's second conjecture: A proof for local hereditary orders in central simple algebras
cites work: Auslander Algebras as Quasi-Hereditary Algebras
cites work: Global dimension two orders are quasi-hereditary
cites work: Finiteness of representation dimension
cites work: Zeta functions and integral representation theory
cites work: Q3869363
description: scientific article; zbMATH DE number 1891314