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 finitedimensional 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 ∞, 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 Mod-S to Mod-R admits an inverse interpretation functor from its image, and hence that, in this case, Mod-R interprets Mod-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 : Mod-R → Mod-S is an interpretation functor such that the smallest definable subcategory containing the image of I is the whole of Mod-S, then if R is tame, so is S and, similarly, if R is domestic, then S also is domestic