A reduction formula for the Cartan determinant problem for algebras
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.
scientific article; zbMATH DE number 446290
scientific article