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An Artin problem for division ring extensions and the pure semisimplicity conjecture. II. (English) Zbl 0953.16012
The author builds on his work in part I [Arch. Math. 66, No. 2, 114-122 (1996; Zbl 0873.16010)] which in turn extends ideas of P. M. Cohn [Proc. Lond. Math. Soc., III. Ser. 11, 531-556 (1961; Zbl 0104.03301)] and A. H. Schofield [Representation of rings over skew fields, Lond. Math. Soc. Lect. Note Ser. 92 (1985; Zbl 0571.16001)] in his efforts to find a counterexample to the pure semisimplicity conjecture (that a ring all of whose modules are direct sums of finitely generated modules must be of finite representation type).
In order to produce a counterexample to this conjecture it would be enough to produce a pair \(F\subseteq G\) of division rings and an \((F,G)\)-bimodule \(M\) such that the dimensions of the modules obtained from \(M\) by successive application of the functors \(\text{Hom}_G(_F(-)_G,G)\) and \(\text{Hom}_F(_F(-)_G,F)\) form an infinite sequence belonging to a certain set that is defined in the paper. The author shows that there are very many, indeed uncountably many, sequences any one of which would, if realized, result in the triangular matrix ring, \(R_M\), formed from \(F\), \(G\) and \(M\) being a counterexample to the conjecture. The author also obtains detailed information on the structure of the module category of any such counterexample \(R_M\).

16G10 Representations of associative Artinian rings
16K40 Infinite-dimensional and general division rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
Full Text: DOI
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