A reduction formula for the Cartan determinant problem for algebras
type: Item
instance of: scholarly article
title: A reduction formula for the Cartan determinant problem for algebras
zbMATH Open document ID: 0797.16009
DOI: 10.1007/BF01258052
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: The Cartan determinant problem is the question whether for a left artinian ring \(A\) of finite global dimension the determinant of the Cartan matrix \(C(A)\) is necessarily 1. The author introduces a matrix \(C_ A(I,A)\), where \(I\) is an ideal of \(A\) and \(_ AI\) has a filtration in a class \(\Delta\) of left \(A\)-modules such that every module in \(\Delta\) is generated by some projective modules in the Grothendieck group \(K_ 0(\text{mod }A)\otimes_{\mathbb{Z}} \mathbb{Q}\) with coefficients in \(\mathbb{Q}\). He proves the reduction formula \(\text{det }C(A/I) = \text{det }C_ A(I,\Delta) \cdot \text{det }C(A)\) and gives applications, especially nice proofs of known theorems concerning the above problem.
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16E20
zbMATH DE Number: 446290
zbMATH Keywords: Cartan determinant problem
zbMATH Keywords: left artinian ring
zbMATH Keywords: finite global dimension
zbMATH Keywords: Cartan matrix
zbMATH Keywords: Grothendieck group
zbMATH Keywords: reduction formula
author: Kunio Yamagata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446290