Document Zbl 1524.18032 - zbMATH Open
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Stratification in tensor triangular geometry with applications to spectral Mackey functors. (English) Zbl 1524.18032
The paper we summarize concerns stratification in the context of tensor triangulated categories. Based on the approach of G. Stevenson [J. Reine Angew. Math. 681, 219–254 (2013; Zbl 1280.18010)] which uses the Balmer-Favi notion of support and the Bamer spectrum (see [P. Balmer and G. Favi, Proc. Lond. Math. Soc. (3) 102, No. 6, 1161–1185 (2011; Zbl 1220.18009); P. Balmer, J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)]), the authors develop, in a first part, the notion of stratification for the rigidly-compactly generated tensor triangulated categories whose Balmer spectrum of compact objects is weakly noetherian. In addition they clarify its relation to the theory of Benson, Iyengar and Krause (“BIK”) (see [D. Benson et al., Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 575–621 (2008; Zbl 1171.18007); Ann. Math. (2) 174, No. 3, 1643–1684 (2011; Zbl 1261.20057); J. Topol. 4, No. 3, 641–666 (2011; Zbl 1239.18013)]).
In the second part, the authors prove that their stratification is equivalent to the classification of localizing \(\otimes\)-ideals and they show that this stratification provides an unified context for many known classification problem, both those that are and are not amenable to the technique of “BIK”. The authors also give and explain the following properties of their stratification namely: universality, permanence and generality.
The authors apply their methods to new examples especially in equivariant homotopy theory. As a special case they obtain a classification of the localizing \(\otimes\)-ideals of the category of derived \(G\)-Mackey functors, \(G\) denotes a finite group.
At the end, the authors consider the telescope conjecture and prove that it holds for every stratified rigidly-compactly generated tensor triangulated categories with generically noetherian spectrum.
MSC:
| 18G80 | Derived categories, triangulated categories |
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 18F99 | Categories in geometry and topology |
| 55P42 | Stable homotopy theory, spectra |
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |