2012 / March Volume 7 No.1
On The Representation Dimension Of Artin Algebras
| Published Date |
2012 / March
|
|---|---|
| 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
|
| 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 $n+2$.
|
| AMS Subject Classification |
35J60, 35B45, 49Q20, 28C99
|
| Received |
2013-10-21
|
| Accepted |
2013-10-21
|