Pure injective envelopes of finite length modules over a generalized Weyl algebra
Generalised Weyl algebras have proved to be a fruitful and important domain of research and a significant source of examples. The Ziegler spectrum of a ring, the space of all isomorphism types of indecomposable pure-injective modules, is coming to be recognized as an important tool in describing the complexity of the category of modules over the ring. The inspiration for the Ziegler spectrum comes from mathematical logic (complexity of first-order definable sets). In this paper the authors investigate the structure of some of the pure-injective modules over generalised Weyl algebras. From the abstract to the paper: ``We consider the pure injective hulls of finite length modules, the elementary duals of these, torsionfree pure injective modules, and the closure in the Ziegler spectrum of the category of finite length modules supported on a non-degenerate orbit of a generalised Weyl algebra. We also show that this category is a direct sum of uniserial categories and admits almost split sequences. We find parallels to but also marked contrast with the behaviour of pure injective modules over finite dimensional algebras and hereditary orders.'' Extensive use is made of the known structure of generalised Weyl algebras and the classification of the simple modules as developed by \textit{V. V. Bavula} [Algebra Anal. 4, No. 1, 75-97 (1992; Zbl 0807.16027), English translation in St. Petersb. Math. J. 4, No. 1, 71-92 (1993)] and by \textit{V. Bavula} and \textit{F. Van Oystaeyen} [J. Algebra 194, No. 2, 521-566 (1997; Zbl 0927.16002)]. The structure theory developed by the authors is quite interesting and I will summarize some of the main results below. The problem of classifying all the indecomposable pure-injective modules over a generalized Weyl algebra \(A\) is unsolvable, so the authors look for likely subfamilies that might be classifiable. One starting point is suggested by the classification of the simple modules: the family of finite length modules supported on a non-degenerate orbit \(O\) of the simple modules. The pure-injective envelope of an indecomposable finite length \(A\)-module \(M\) is indecomposable, and is an extension of \(M\) by a direct sum of copies of the quotient skew field of \(A\). Every finite length module supported on \(O\) is a direct sum of uniserial homogeneous modules, and the category of all such admits almost split sequences. The direct limit along a natural ray of monomorphisms of these modules leads to a ``Prüfer''-type module; this module is not, as we might hope, pure-injective, but its pure-injective envelope is indecomposable. The corresponding inverse limit construction of an ``adic''-type module produces a module whose pure-injective envelope has a complicated direct sum decomposition, but nonetheless all the components lie in the Ziegler spectrum of the orbit \(O\). The Cantor-Bendixson rank of this spectrum is 2, with the unique generic point being the quotient skew field of \(A\). The authors are also able to describe all the indecomposable pure-injective torsionfree modules over \(A\). The authors conclude with a comparison of some standard dualities in this setting, and a brief consideration of the case of degenerate orbits.
scientific article; zbMATH DE number 1797467
scientific article