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.
No background knowledge in logic or model theory will be assumed.