Decidability of theories of modules over tubular algebras
arXiv:1603.03284 · doi:10.1112/plms.12403
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.
The results have been extended to cover all tubular algebras rather than just canonical algebras of tubular type - the title and abstract have been changed to reflect this. Moreover, the counter example to corollary 8.8 of Prest and Harland's paper "Modules with irrational slope over tubular algebras" has be replaced by a proof that that corollary does not hold for any tubular algebra. 47 pages