The spectrum of prime ideals in tensor triangulated categories
Abstract
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.
Walter de Gruyter GmbH & Co. KG
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- Semilattices of groups and inductive limits of Cuntz algebras
- Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields
- Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli
- Arcs, valuations and the Nash map
- On Artin’s L-functions. II: Dirichlet coefficients
- A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces
- The spectrum of prime ideals in tensor triangulated categories
- Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes
- On the ‘Section Conjecture’ in anabelian geometry
Articles in the same Issue
- Semilattices of groups and inductive limits of Cuntz algebras
- Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields
- Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli
- Arcs, valuations and the Nash map
- On Artin’s L-functions. II: Dirichlet coefficients
- A complex ball uniformization of the moduli space of cubic surfaces via periods of K 3 surfaces
- The spectrum of prime ideals in tensor triangulated categories
- Characterization of the linear partial di¤erential equations that admit solution operators on Gevrey classes
- On the ‘Section Conjecture’ in anabelian geometry
