Pure semisimplicity conjecture and Artin problem for dimension sequences

TL;DR

This paper explores the pure semisimplicity conjecture by linking it to the existence of certain hereditary artinian rings and tight embeddings of division rings, proposing a conjecture that could disprove the conjecture.

Contribution

It establishes a connection between the pure semisimplicity conjecture and the existence of tight embeddings of division rings, introducing Conjecture A as a potential disproof.

Findings

Existence of counterexamples would follow from certain hereditary artinian rings.

Such rings are equivalent to specific tight embeddings of division rings.

A new division ring extension with a particular dimension sequence is constructed.

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.