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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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