Representation Type, Decidability and Pseudofinite-dimensional Modules over Finite-dimensional Algebras

Abstract: 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 (theory of) a class of modules is said to be decidable if there is an algorithm which given a sentence in the language of modules (a sentence is a particular kind of statement about modules) answers whether it is true in all modules in that class. A long-standing conjecture of Mike Prest claims that the (theory of) the class of all modules over a finite-dimensional algebra is decidable theory if and only if it is of tame representation type. The reverse direction of this conjecture is often hard to prove even in particular examples. One difficulty is that the conjecture talks about all modules rather than just finite-dimensional ones. In this talk I will present work in progress around and in support of a new conjecture, inspired by Prest’s conjecture, which claims that the (theory of) the class of finite-dimensional modules over a finite-dimensional algebra is decidable if and only if it is of tame representation type.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.