Lorna Gregory (University of East Anglia) – Representation Type, Decidability and Pseudofinite-dimensional Modules over Finite-dimensional Algebras

The representation type of a finite-dimensional k-algebra is an algebraic measure of how hard it is to classify its finite-dimensional indecomposable modules.

Intuitively, a finite-dimensional k-algebra is of tame representation type if we can classify its finite-dimensional modules and wild representation type if its module category contains a copy of the category of finite-dimensional modules of all other finite-dimensional k-algebras. An archetypical (although not finite-dimensional) tame algebra is k[x]. The structure theorem for finitely generated modules over a PID describes its finite-dimensional modules. Drozd’s famous dichotomy theorem states that all finite-dimensional algebras are either wild or tame.

The tame/wild dividing line is not seen by standard model theoretic invariants or even the more specialised invariants coming from Model Theory of Modules. A long-standing conjecture of Mike Prest claims that a finite-dimensional algebra has decidable theory of modules if and only if it is of tame representation type. More recently, I conjectured that a finite-dimensional algebra has decidable theory of (pseudo)finite dimensional modules if and only if it is of tame representation type. This talk will focus on recent work providing evidence for the second conjecture.