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Journal für die reine und angewandte Mathematik

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Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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The spectrum of prime ideals in tensor triangulated categories

Paul Balmer

Citation Information: Journal für die reine und angewandte Mathematik. Volume 2005, Issue 588, Pages 149–168, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crll.2005.2005.588.149, December 2005

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We define the spectrum  of a tensor triangulated category 

as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports  for objects of
. This construction is functorial with respect to all tensor triangulated functors. Several elementary properties of schemes hold for such spaces, e.g. the existence of generic points or some quasi-compactness. Locally trivial morphisms are proved to be nilpotent. We establish in complete generality a classification of thick ⊗-ideal subcategories in terms of arbitrary unions of closed subsets with quasi-compact complements (Thomason’s theorem for schemes, mutatis mutandis). We also equip this spectrum with a sheaf of rings, turning it into a locally ringed space. We compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory.

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