The Zariski spectrum of the category of finitely presented modules

Prest, Mike (2008) The Zariski spectrum of the category of finitely presented modules. [MIMS Preprint]

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Abstract

A representation-theoretic description of the Zariski spectrum of a commutative noetherian ring is applied to more general categories, giving the "Gabriel-Zariski" spectrum. Applied to functor categories it gives a topology, the "rep-Zariski spectrum" on the set of indecomposable pure-injective modules. This topology is dual to Ziegler's topology on the same underlying set. Associated presheaves of rings and of small abelian categories are defined. Examples of rep-Zariski spectra are computed. Over commutative coherent rings it is shown that, although its underlying set might be larger, the Gabriel-Zariski spectrum is topologically equivalent to the Zariski spectrum.

Item Type: MIMS Preprint
Additional Information: This is the final version of this paper. I expect not to submit this for publication (much of the content will appear elsewhere).
Uncontrolled Keywords: Zariski spectrum, injective, pure-injective, Gabriel-Zariski spectrum, rep-Zariski spectrum, Ziegler spectrum, structure sheaf, ring of definable scalars, category of definable scalars, finite type localisation, abelian category, commutative coherent ring.
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations
MSC 2010, the AMS's Mathematics Subject Classification > 16 Associative rings and algebras
MSC 2010, the AMS's Mathematics Subject Classification > 18 Category theory; homological algebra
Depositing User: Professor Mike Prest
Date Deposited: 05 Mar 2008
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/1049

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