Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings. (English) Zbl 1204.18005

Balmer, Paul

Algebr. Geom. Topol. 10, No. 3, 1521-1563 (2010).

The author [P. Balmer, J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)] has introduced the notion of a triangular spectrum \(\text{Spc}(\mathcal{K})\) for a tensor triangulated category \(\mathcal{K}\). From the previous work of the author and others it is clear that \(\text{Spc}(\mathcal{K})\) is a natural home for triangular geometry. However, computing this spectrum is a very hard problem. It is essentially equivalent to the problem of classifying the tensor thick ideals of \(\mathcal{K}\). The latter is a very rich topic and has been studied in the fields of stable homotopy theory, homological algebra, algebraic geometry and modular representation theory. Therefore \(\text{Spc}(\mathcal{K})\) is a natural and interesting object of study.
In the paper under review the author constructs a natural and continuous map
\[ \rho_{\mathcal{K}} : \text{Spc}(\mathcal{K}) \rightarrow \text{Spec} (\text{End}_{\mathcal{K}}(\mathbb{I})) \]
from the triangular spectrum \(\text{Spc}(\mathcal{K})\) to the Zariski spectrum \(\text{Spec} (\text{End}_{\mathcal{K}}(\mathbb{I}))\) of the endomorphism ring of the unit object \(\mathbb{I}\) in the tensor triangulated category \(\mathcal{K}\). It is shown that this map is often surjective but far from injective in general. For instance, it is shown that when \(\mathcal{K}\) is connective, i.e., \(\text{Hom}_{\mathcal{K}}(\Sigma^{i}(\mathbb{I}), \mathbb{I}) = 0\) for \(i < 0\), then \(\rho_{\mathcal{K}}\) is surjective. This applies to derived categories of rings and also to the stable homotopy category of finite spectra. Graded version of this result is also proved in Theorem 7.3 and Corollary 7.4. Using these results the author is able to shed some light on \(\text{Spc}(\mathcal{K})\) in examples coming from \(\mathbb{A}^{1}\)-homotopy theory and non-commutative topology where \(\text{Spc}(\mathcal{K})\) has not been known; see Corollary 10.1 and Corollary 8.8.

Reviewer: Sunil K. Chebolu (Normal, IL)

MSC:

18E30	Derived categories, triangulated categories (MSC2010)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
19K35	Kasparov theory (\(KK\)-theory)
20C20	Modular representations and characters
55P42	Stable homotopy theory, spectra
55U35	Abstract and axiomatic homotopy theory in algebraic topology
18G55	Nonabelian homotopical algebra (MSC2010)
18G30	Simplicial sets; simplicial objects in a category (MSC2010)

Keywords:

tensor triangular geometry; spectra

Citations:

Zbl 1080.18007

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