On The Representation Dimension Of Artin Algebras
by Claus Michael Ringel  

Vol. 7 No. 1 (2012) P.33~P.70

ABSTRACT

The representation dimension of an artin algebra as introduced by M.~Auslander in his Queen Mary Notes is the minimal possible global dimension of the endomorphism ring of a generator-cogenerator. The following report is based on two texts written in 2008 in connection with a workshop at Bielefeld. The first part presents a full proof that any torsionless-finite artin algebra has representation dimension at most $3$, and provides a long list of classes of algebras which are torsionless-finite. In the second part we show that the representation dimension is adjusted very well to forming tensor products of algebras. In this way one obtains a wealth of examples of artin algebras with large representation dimension. In particular, we show: The tensor product of $n$ representation-infinite path algebras of bipartite quivers has representation dimension precisely $n+2$.

KEYWORDS
Artin algebras, representation dimension, torsionless modules, divisible modules, torsionless-finite algebras, minimal representation-infinite algebras, special biserial algebras, Oppermann dimension, lattices, tensor products of algebras, tiered algebras, bipartite quivers

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 16G10, 16G20, 16G60, 16D90, 16E10, 16G30

MILESTONES

Received: 2011-06-14
Revised : 2011-09-12
Accepted: 2011-09-13

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