Intersecting the dimension and slice filtrations for motives
type: Item
instance of: scholarly article
title: Intersecting the dimension and slice filtrations for motives
zbMATH Open document ID: 1394.14005
published in: Homology, Homotopy and Applications
publication date: 2018-03-29T00:00:00Z
review text: Let \(DM^{\mathrm{eff}}_{R}\) be the monoidal triangulated category of unbounded \(R\)-linear motivic complexes over a perfect field \(k.\) A ring \(R\) is a coefficient ring such that \({\mathrm{char}}\,k\) is invertible if it is positive. Let \(d_{\leq m}DM^{\mathrm{eff}}_{R}\) be the localising subcategory of \(DM^{\mathrm{eff}}_{R}\) generated by the motives of smooth varieties of dimension at most \(m.\) Let \(R\langle i\rangle\) be the Lefschetz motif. The category \(d_{\leq m}DM^{\mathrm{eff}}_{R}\) contains \(d_{\leq m-i}DM^{\mathrm{eff}}_{R}{\otimes}_{d_{DM^{\mathrm{eff}}_{R}} }R\langle i\rangle \) and the functor \(-\langle i\rangle = - {\otimes}_{{DM^{\mathrm{eff}}_{R}} }R\langle i\rangle : M \rightarrow M\langle i\rangle\) is a full embedding of \(DM^{\mathrm{eff}}_{R}\) into itself [\textit{A. Beilinson} and \textit{V. Vologodsky}, Geom. Funct. Anal. 17, No. 6, 1709--1787 (2008; Zbl 1144.14014)]. The question that the author considers is whether the intersection of \(d_{\leq m}DM^{\mathrm{eff}}_{R}\) with \(DM^{\mathrm{eff}}_{R}\langle i\rangle\) ( the \(i\)-th level of the slice filtration of \(DM^{\mathrm{eff}}_{R}\) ) is as small as possible i.e. equals \(d_{\leq m-i}DM^{\mathrm{eff}}_{R}\langle i\rangle.\) The author proves the following equality \[ d_{\leq m}DM^{\mathrm{eff}}_{-,R} \cap DM^{\mathrm{eff}}_{-,R}\langle i\rangle = d_{\leq m-i}DM^{\mathrm{eff}}_{-,R}\langle i\rangle, \] where \(DM^{\mathrm{eff}}_{-,R}\) is the \(R\)-linear version of the category of bounded above motivic complexes. This is applied to obtain a reformulation of a conjecture of \textit{J. Ayoub} [in: Hodge Theory and \(L^2\)-analysis. Somerville, MA: International Press; Beijing: Higher Education Press. 87--125 (2017; Zbl 1379.14003)].
reviewed by: Piotr Krasoń
Mathematics Subject Classification ID: 14C15
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 19E15
zbMATH DE Number: 6854560
zbMATH Keywords: Voevodsky motive
zbMATH Keywords: dimension filtration
zbMATH Keywords: slice filtration
zbMATH Keywords: weight structure
zbMATH Keywords: triangulated category
zbMATH Keywords: Chow motive
author: Mikhail V. Bondarko
MaRDI profile type: Publication
DOI: 10.4310/HHA.2018.V20.N1.A16
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description: scientific article; zbMATH DE number 6854560