The representation dimension of k [ x, y ] / ( x², yⁿ )

verfasst von

Thorsten Holm, Wei Hu

Abstract

The representation dimension of an Artin algebra was defined by M. Auslander in 1970. The precise value is not known in general, and is very hard to compute even for small examples. For group algebras, it is known in the case of cyclic Sylow subgroups. For some group algebras (in characteristic 2) of rank at least 3 the precise value of the representation dimension follows from recent work of R. Rouquier. There is a gap for group algebras of rank 2. In this paper we show that for all n {greater than or slanted equal to} 0 and any field k the commutative algebras k [ x, y ] / ( x², y^{2 + n} ) have representation dimension 3. For the proof, we give an explicit inductive construction of a suitable generator-cogenerator. As a consequence, we obtain that the group algebras in characteristic 2 of the groups C₂ × C₂m ^{have representation dimension 3. Note that for m {greater than or slanted equal to} 3 these group algebras have wild representation type.}

Externe Organisation(en)

University of Leeds
Beijing Normal University

Typ

Artikel

Journal

Journal of algebra

Band

301

Seiten

791-802

Anzahl der Seiten

12

ISSN

0021-8693

Publikationsdatum

15.07.2006

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie

Elektronische Version(en)

https://doi.org/10.1016/j.jalgebra.2005.11.037 (Zugang: Offen)