The spectrum of prime ideals in tensor triangulated categories

By Paul Balmer. 2004

386 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd4bac0baa3b18d51568e124fc7fefb02d57aeec

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In Mathematical Physics

Citations in the corpus

Support varieties: an ideal approach (2005, methodology): “and call this subset of Spec C the support of x. It is convenient to work with support data defined on objects of C instead of support data defined on ideals of C. This motivates the following definition from [ 1 ].”
Spectra, spectra, spectra – Tensor triangular spectra versus Zariski spectra of endomorphism rings (2010, methodology): “Somewhat independent of the rest, Section 6 recalls from [1] the sheaf of rings OK on the spectrum Spc.”
Generalized tensor idempotents and the telescope conjecture (2011, background): “Tensor idempotents and geometry We quickly recall elements of tensor triangular geometry from [2, 3, 7].”
The spectrum of the equivariant stable homotopy category of a finite group (2015, background): “Proof It is basic tt-geometry [1] that every irreducible closed subset of Spc(K) has a unique generic point and that the closure of a point P is exactly { Q ∈ Spc(K)∣Q ⊆ P; beware the reversal of the inclusion compared to the Zariski topology.”
The homotopy category of flat modules, and Grothendieck duality (2008, background): “The category K(R-Flat) is a tensor triangulated category, in the terminology of Balmer [1,2].”
Separability and triangulated categories (2011, background): “In terms of tensor triangular geometry [3], this opens the way to étale morphisms of tensor triangulated categories, extending the theory of [2] beyond the Zariski topology.”
Classifying thick subcategories of the stable category of Cohen-Macaulay modules (2009, neutral): “For example, see [7, 8, 9, 43, 15, 11, 38, 13, 20, 53].”
Supports and filtrations in algebraic geometry and modular representation theory (2007, methodology): “Results from [1] which we use only once in this paper will be quoted when needed.”
Subcategories of singularity categories via tensor actions (2011, methodology): “Given a specialization closed subset V ⊆ Spc T c we denote by T c V the thick subcategory of compact objects supported (in the sense of [5]) on V .”