Decidability of theories of modules over tubular algebras

TL;DR

This paper proves that the common theory of modules over tubular algebras is decidable, providing evidence for a conjecture linking tameness of algebras to decidability of their module theories, and confirms this for certain canonical algebras.

Contribution

It establishes the decidability of module theories over tubular and concealed canonical algebras, supporting a conjecture relating tameness and decidability.

Findings

Decidable theory for modules over tubular algebras.

Confirmation of Prest's conjecture for concealed canonical algebras.

First examples of non-domestic algebras with decidable module theories.

Abstract

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra (over a recursively given field) is tame if and only its common theory of modules is decidable. Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. These are the first examples of non-domestic algebras which have been shown to have decidable theory of modules.