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Direct products of modules and the pure semisimplicity conjecture. Part II

Published online by Cambridge University Press:  25 July 2002

Birge Huisgen-Zimmermann  and
Manuel Saorín
Birge Huisgen-Zimmermann
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA e-mail: [email protected]
Manuel Saorín
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo-MU, Spain e-mail: [email protected]
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Abstract

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We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 44 , Issue 2 , May 2002 , pp. 317 - 321
Copyright
2002 Glasgow Mathematical Journal Trust