The Ziegler Spectrum of a Locally Coherent Grothendieck Category

Abstract

The general theory of locally coherent Grothendieck categories is presented. To each locally coherent Grothendieck category C a topological space, the Ziegler spectrum of C, is associated. It is proved that the open subsets of the Ziegler spectrum of C are in bijective correspondence with the Serre subcategories of coh C the subcategory of coherent objects of C. This is a Nullstellensatz for locally coherent Grothendieck categories. If R is a ring, there is a canonical locally coherent Grothendieck category RC (respectively, CR) used for the study of left (respectively, right) R-modules. This category contains the category of R-modules and its Ziegler spectrum is quasi-compact, a property used to construct large (not finitely generated) indecomposable modules over an artin algebra. Two kinds of examples of locally coherent Grothendieck categories are given: the abstract category theoretic examples arising from torsion and localization and the examples that arise from particular modules over the ring R. The duality between coh-(RC) and coh-CR is shown to give an isomorphism between the topologies of the left and right Ziegler spectra of a ring R. The Nullstellensatz is used to give a proof of the result of Crawley-Boevey that every character ξ: K0(coh-C) → Z is uniquely expressible as a Z-linear combination of irreducible characters. 1991 Mathematics Subject Classification: 16D90, 18E15.