Ziegler spectra of tame hereditary algebras.

*(English)*Zbl 0936.16014Let \(A\) be a tame hereditary finite dimensional algebra over a field. The points of the Ziegler spectrum of \(A\), the indecomposable pure-injective modules, are well-known. The paper under review describes the topological structure of the Ziegler spectrum. The proofs are based on a localization technique introduced by W. W. Crawley-Boevey [Proc. Lond. Math. Soc., III. Ser. 62, No. 3, 490-508 (1991; Zbl 0768.16003)] combined with a description of the Ziegler spectrum of PI Dedekind domains, obtained in the first part of the paper. Similar results have been obtained by C. M. Ringel [Colloq. Math. 76, No. 1, 105-115 (1998; Zbl 0901.16006)], whose approach is different.

The following remark has been submitted by the author: ‘It has been pointed out to me (Mike Prest) by G. Puninski that the last paragraph of page 162 is wrong. The first assertion made, “that every finitely presented module embeds purely in a direct product of torsion modules” is correct, but the remainder of that paragraph, “and hence” onwards, should be deleted (since the argument and the conclusion are incorrect)’.

The following remark has been submitted by the author: ‘It has been pointed out to me (Mike Prest) by G. Puninski that the last paragraph of page 162 is wrong. The first assertion made, “that every finitely presented module embeds purely in a direct product of torsion modules” is correct, but the remainder of that paragraph, “and hence” onwards, should be deleted (since the argument and the conclusion are incorrect)’.

Reviewer: Steffen König (Bielefeld)

##### MSC:

16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |

16D50 | Injective modules, self-injective associative rings |

16G10 | Representations of associative Artinian rings |

16B70 | Applications of logic in associative algebras |

03C60 | Model-theoretic algebra |