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The Ziegler and Zariski spectra of some domestic string algebras. (English) Zbl 1061.16022
Summary: It was a conjecture of the second author that the Cantor-Bendixson rank of the Ziegler spectrum of a finite-dimensional algebra is either less than or equal to 2 or is undefined. Here we refute this conjecture by describing the Ziegler spectra of some domestic string algebras where arbitrary finite values greater than 2 are obtained. We give a complete description of the Ziegler and Gabriel-Zariski spectra of the simplest of these algebras. The conjecture has been independently refuted by J. Schröer who, extending his work [Hammocks for string algebras, Dissertation, Univ. Bielefeld, Bielefeld (1997; Zbl 0926.16015)] on these algebras, computed their Krull-Gabriel dimension [Math. Z. 233, No. 2, 287-303 (2000; Zbl 0960.16011)].

MSC:
16G20 Representations of quivers and partially ordered sets
03C60 Model-theoretic algebra
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
18E15 Grothendieck categories (MSC2010)
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