Christian Lomp

In 1960, Eben Matlis showed that the injective hull of a simple module over a commutative Noetherian ring is Artinian. Matlis’ result does not extend to arbitrary non-commutative Noetherian rings. 

However, there are some interesting classes of Noetherian KK-algebras for which the injective hull of finite-dimensional simple modules still satisfies some finiteness conditions, such as being locally finite-dimensional. 

Our study was motivated by various results in the literature: Let G be a polycyclic-by-finite group and K[G] its group ring over a field K. 

In 1982, Stephen Donkin showed that any injective hull of a finite-dimensional K[G]-module is locally finite-dimensional (Donkin attributes this result to Ken Brown).

 Interest in this question actually dates back to works by Philip Hall and J. E. Roseblade from the 1960s and 1970s on finitely generated soluble groups. 

In this talk, based on a joint paper with Can Hatipoglu, we provide necessary and sufficient conditions for the category of locally finite-dimensional representations over a Noetherian algebra to be closed under taking injective hulls. 

We also extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products, and affine Hopf algebras of low Gelfand-Kirillov dimension.