Representation embeddings, interpretation functors and controlled wild algebras

Lorna Gregory, Mike Prest

Published 2014-11-12Version 1

We show that representation embeddings between categories of representations 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 representations of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}{-}S$ to ${\rm Mod}{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}{-}R$ interprets ${\rm Mod}{-}S$. 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.