Decidability of the theory of modules over Pr\"ufer domains with infinite residue fields

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Pr\"ufer (in particular B\'ezout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For B\'{e}zout domains these conditions are also necessary.Comment: Updated so that the title and abstract matches the published version. Other minor corrections and changes mad

Representation embeddings, interpretation functors and controlled wild algebras

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from {\rm Mod}\mbox{-}S to {\rm Mod}\mbox{-}R admits an inverse interpretation functor from its image and hence that, in this case, {\rm Mod}\mbox{-}R interprets {\rm Mod}\mbox{-}S. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of {\rm Mod}\mbox{-}S then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.Comment: More results adde

Maranda’s theorem for pure-injective modules and duality

Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by π Let Λ be an R-order such that QΛ is a separable Q-algebra.Maranda showed that there exists k ∈ N such that for all Λ-lattices L and M, if L/L πk ≈ then L ≈ M Moreover, if R is complete and L is an indecomposable Λ-lattice, then L/L πk is also indecomposable. We extend Marandafs theorem to the class of R-reduced R-torsion-free pure-injective Λ-modules. As an application of this extension,we showthat if Λis an order over a Dedekind domain R with field of fractions Q such that QΛ is separable then the lattice of open subsets of the R-torsion-free part of the right Ziegler spectrum of Λ is isomorphic to the lattice of open subsets of the R-torsion-free part of the left Ziegler spectrum of Λ. Further, with k as in Maranda'fs theorem, we show that if M is R-torsion-free and H(M) is the pureinjective hull of M then H(M)/H(M) πk is the pure-injective hull of M/Mπk. We use this result to give a characterisation of R-torsion-free pure-injective Λ-modules and describe the pure-injective hulls of certain R-torsion-free Λ-modules

Decidability of theories of modules over tubular algebras

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 suitably recursive field) is tame if and only if its common theory of modules is decidable (Prest, Model theory and modules (Cambridge University Press, Cambridge, 1988)). Moreover, as a corollary, we are able to confirm this conjecture for the class of concealed canonical algebras over algebraically closed fields. Tubular algebras are the first examples of non-domestic algebras which have been shown to have decidable theory of modules. We also correct results in Harland and Prest (Proc. Lond. Math. Soc. (3) 110 (2015) 695–720), in particular, Corollary 8.8 of that paper