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2006.107: The Zariski spectrum of the category of finitely presented modules

2006.107: Mike Prest (2008) The Zariski spectrum of the category of finitely presented modules.

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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 2000 > 03 Mathematical logic and foundations
MSC 2000 > 16 Associative rings and algebras
MSC 2000 > 18 Category theory; homological algebra
MIMS number:2006.107
Deposited By:Professor Mike Prest
Deposited On:05 March 2008

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