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The Ziegler spectrum of a locally coherent Grothendieck category. (English) Zbl 0909.16004
The author studies categorical versions of model theoretic methods for modules, especially categorical versions of the results by M. Ziegler [Ann. Pure Appl. Logic 26, 149-213 (1984; Zbl 0593.16019)]. The main results are in Section 3 where the author defines a topological space, called the Ziegler spectrum, for a locally coherent Grothendieck category. This generalizes the Ziegler spectrum for a ring originally defined by Ziegler. The main result is a “Nullstellensatz”, Theorem (3.8): for a locally coherent Grothendieck category \(\mathcal C\) it states the existence of an inclusion preserving bijective correspondence between the Serre subcategories of \(\mathcal C\) and the open subsets \(\mathcal O\) of the Ziegler spectrum \(Zg({\mathcal C})\). The following sections show examples and applications to the study of categories of modules over a ring \(R\). The author applies the methods developed in this paper to treat recent results by W. W. Crawley-Boevey [in Representations of algebras and related topics, Lond. Math. Soc. Lect. Note Ser. 168, 127-184 (1992; Zbl 0805.16028)]. Crawley-Boevey has shown that for an infinite Artin algebra \(\Lambda\) the Second Brauer-Thrall Conjecture is equivalent to the existence of a non-isolated endofinite point (i.e. of finite length over their endomorphism ring) in Ziegler’s spectrum \(Zg(_\Lambda{\mathcal C})\).
Examples of Serre subcategories are used in the analysis of the categories of modules. The last two sections study the Grothendieck group \(K_0(\text{coh }{\mathcal C})\) and its characters.

MSC:
16D90 Module categories in associative algebras
18E15 Grothendieck categories (MSC2010)
03C60 Model-theoretic algebra
16G10 Representations of associative Artinian rings
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