Colloquium Mathematicum 77.1, 1998
pp. 121 - 132
The Local Duality for Homomorphisms and an Application to Pure Semisimple PI-Rings
Abstract.
The local duality L is a useful tool both in module theory and in representation theory. For example, it is a key ingredient in the construction of Auslander-Reiten sequences for finitely presented modules. If k is a commutative artinian ring and R a k -algebra, the local duality coincides on the finite length modules with the functorial duality D which is given by the injective hull of the radical factor of k .
The local duality L is not functorial in general. It is the aim of this article to show that L has the following related properties.
- The local duality L commutes with finite direct sums, up to isomorphism, provided each summand has perfect endomorphism ring.
- The local duality can be defined for homomorphisms between R modules and it behaves well on a class of homomorphisms which we call "endofinite". However, this class may not be closed under addition or composition.
- For artinian right pure semisimple PI-rings, the local duality induces L a bijection between the isoclasses of indecomposable finite length left and right modules. We use this bijection to obtain a new proof for the fact proved by Herzog that such rings are of finite representation type.
Mathematical Reviews: 99f:16009
Zentralblatt: 915.16001
Last modified: October 12, 1999 by Markus Schmidmeier