Auslander-Reiten sequences over derivation polynomial rings

Abstract

Let F be a field with derivation ξ → ξ' and R = F[ Y,'] the corresponding derivation polynomial ring. In this article we provide necessary and sufficient conditions for the existence of enough Auslander-Reiten sequences in the category m R of R-modules of finite length. Since in some respects R-modules can be treated in the same way like modules over a group algebra, only the existence of an Auslander-Reiten sequence in m R ending with the simple module R YR has to be explored. Furthermore, we show that certain generalized power series fields F satisfy the characteristic conditions of the existence theorem: a typical example is the field k(( X)) of formal Laurent series over a field k of characteristic 0.