Representation embeddings and the third Brauer--Thrall conjecture

Abstract

Let $A$ be a representation-infinite finite-dimensional algebra over an algebraically closed field $k$. We prove that for every infinite cardinal $κ$ there is an indecomposable $A$-module $M$ with $\dim_k M=κ$. Over algebraically closed fields, this settles Simson's conjecture, also called the third Brauer--Thrall conjecture, that every representation-infinite finite-dimensional algebra has indecomposable modules of arbitrarily large cardinality. To construct these modules, we extend Bongartz's representation embedding from finite-dimensional Kronecker modules without a simple injective summand to such modules of arbitrary dimension. The extended functor commutes with direct sums and filtered colimits and preserves the dimension of infinite-dimensional modules. Applying it to the indecomposable Kronecker modules of every infinite dimension constructed by Göbel and Simson yields the required $A$-modules.