## 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.*

*This is the latest version of this eprint.*

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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 |
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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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