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.
Highlights
If R is a ring we write mod-R for the category of finitely presented right R-modules, Mod-R for the category of all right R-modules and ind-R for the set of isomorphism classes of finitely presented indecomposable right R-modules
We will usually assume that finite-dimensional algebras are basic, connected and over an algebraically closed base field
That every finite-dimensional algebra is (k-linearly) Morita equivalent to a basic algebra and every basic finite-dimensional algebra is isomorphic to a finite product of basic connected algebras
Summary
If R is a ring we write mod-R for the category of finitely presented right R-modules, Mod-R for the category of all right R-modules and ind-R for the set of isomorphism classes of finitely presented indecomposable right R-modules. By. given a finite-dimensional algebra R over an algebraically closed recursive field k, in order to show that the theory of R-modules is decidable it is enough to show that there is an algorithm which, given a boolean combination χ of invariant sentences of the form |φ/ψ| > 1, answers whether there is an R-module in which χ is true. I : Mod-R → Mod-S, is specified (up to equivalence) by giving a pp-m-pair φ/ψ and, for each s ∈ S, a pp-2m-formula ρs such that, for all M ∈ Mod-R, the solution set ρs(M, M ) ⊆ M m × M m defines an endomorphism of φ(M )/ψ(M ) as an abelian group, and such that φ(M )/ψ(M ), together with the ρs actions, is an S-module (see [Pre97] or [Pre09, 18.2.1]). If k is a recursive field the induced mapping on sentences in the previous paragraph is effective
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