2012 / March Volume 7 No.1
On The Representation Dimension Of Artin Algebras
Published Date |
2012 / March
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Title | On The Representation Dimension Of Artin Algebras |
Author | |
Keyword |
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, 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, Optimal transportation, obliqueness, Monge-Ampère equation
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Download | |
Pagination | 33-70 |
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 $n+2$.
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AMS Subject Classification |
35J60, 35B45, 49Q20, 28C99
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Received |
2013-10-21
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Accepted |
2013-10-21
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