Pure semisimplicity conjecture and Artin problem for dimension sequences

Abstract

Inspired by a recent paper due to Jos\'{e} Luis Garc\'{i}a, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding $F\hookrightarrow M_n(G)$ exists provided that $n<5$. As a byproduct, we obtain a division ring extension $G\subseteq F$ such that the bimodule ${}_GF_F$ has the right dimension sequence $(1,2,2,2,1,4)$. Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture.