On stratification for spaces with Noetherian mod \(p\) cohomology. (English) Zbl 1505.55022
Let \(X\) be a topological space with Noetherian mod \(p\) cohomology and let \(C^*(X;\mathbb{F}_p)\) be the commutative ring spectrum of \(\mathbb{F}_p\)-valued cochains on \(X\). This paper aims to exhibit conditions under which the category of module spectra on \(C^*(X;\mathbb{F}_p)\) is stratified in the sense of Benson, Iyengar, Krause (cf. [D. J. Benson et al., Ann. Math. (2) 174, No. 3, 1643–1684 (2011; Zbl 1261.20057)]), providing a classification of all its localizing subcategories. The authors consider some families of naturally occurring examples, including the classifying spaces of Kac-Moody groups and spaces \(X\) that admit an \(H\)-space structure. For these families they show that the category of module spectra over the corresponding cochain ring spectrum is stratified. Furthemore, they consider spaces that are \(p\)-good in the sense of A. K. Bousfield and D. M. Kan [Homotopy limits, completions and localizations. Springer, Cham (1972; Zbl 0259.55004)], and with Noetherian mod-\(p\) cohomology. For such spaces they show that the existence of a stratification for the corresponding categories of module spectra is equivalent to a condition that generalizes Chouinard’s theorem for finite groups in a homotopy theoretic context [L. G. Chouinard, J. Pure Appl. Algebra 7, 287–302 (1976; Zbl 0327.20020)]. In particular, using the work of Benson-Iyengar-Krause [D. Benson et al., Ann. Sci. Éc. Norm. Supér. (4) 41, No. 4, 575–621 (2008; Zbl 1171.18007)], this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory. The paper is very clearly written and is a remarkable example of a beautiful interaction between stable and unstable homotopy theory.
Reviewer: Ran Levi (Aberdeen)
MSC:
| 55P42 | Stable homotopy theory, spectra |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
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