Intersecting the dimension and slice filtrations for motives
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)].
scientific article; zbMATH DE number 6854560
scientific article