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The homotopy Leray spectral sequence. (English) Zbl 1440.14122
Binda, Federico (ed.) et al., Motivic homotopy theory and refined enumerative geometry. Workshop, Universität Duisburg-Essen, Essen, Germany, May 14–18, 2018. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 745, 21-68 (2020).
The authors define and study the homotopy Leray spectral sequence in motivic homotopy theory, in light of the Leray spectral sequence in topology, which generalizes M. Rost’s spectral sequence for cycle modules [Doc. Math. 1, 319–393 (1996; Zbl 0864.14002)]. Let \(f:X\to B\) be a morphism of schemes and let \(\pi:X\to S\) be a separated morphism essentially of finite type. Let \(\delta\) be a dimension function on \(S\) and let \(\mathbb{E}\) be a motivic spectrum on \(S\). The main result (Theorem 4.1.2) shows the existence of a spectral sequence of the form \[ E^2_{p,q}=H_p(B,H^\delta_q(f_*\pi^!\mathbb{E}))\Longrightarrow \mathbb{E}_{p+q}(X/S) \] where \(H^\delta_q\) is the fiber \(\delta\)-homology (Definition 3.1.2), and \(\mathbb{E}_{p+q}\) is the bivariant theory (see Notations and Conventions). The filtration of the abutment agrees with the \(\delta\)-niveau filtration (Proposition 4.1.9), and it is expected that the spectral sequence is closely related to the \(\delta\)-niveau spectral sequence [M. Bondarko and F. Déglise, Adv. Math. 311, 91–189 (2017; Zbl 1403.14053)]). A version for cohomology theories instead of bivariant theories is defined in Theorem 4.2.5.
Reviewer’s remark: Note that the proof of the existence of the functor \(f_!\) for morphisms essentially of finite type in Proposition 4.2.6 is missing, see Appendix B of F. Déglise et al., “On the rational motivic homotopy category”, Preprint, arXiv:2005.10147] for a discussion.
For the entire collection see [Zbl 1435.14021].
MSC:
14F42 Motivic cohomology; motivic homotopy theory
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
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