On Chow weight structures for \(cdh\)-motives with integral coefficients (Q2822780)

\noindent Let \(S\) be an excellent finite dimensional Noetherian scheme of characteristic \(p\) (a prime or zero) and \(R\) a commutative associative coefficient ring with unit. For a prime \(p\) assume that \(p\) is invertible in \(R.\) The paper is devoted to defining a Chow weight structure on the category \({\mathcal DM}_{c}(S)\) of constructible \(cdh\)-motives over an equicharacteristic scheme \(S.\) This is a part of a program of lifting Deligne's weights for étale sheaves and mixed Hodge structures to motives. The first author defined in [\textit{M. V. Bondarko}, J. K-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019)] the Chow weight structrure on the triangulated category of Voevodsky's motives \({\mathcal DM}_{gm}\) (with integral coefficients) over a field of characteristic \(0.\) In this case the heart of this weight structure is the classical category of Chow motives. This approach is alternative to the conjectural Chow-Kunneth decompositions. In the paper the authors define Chow weight structure on \({\mathbb Z}[\frac{1}{p}]\)-linear motives or more generally on \(R\)-linear motives for any \({\mathbb Z}[\frac{1}{p}]\)-algebra \(R\) over \(S.\) In the case \(p=0\) the authors substitute \({\mathbb Z}\) for \({\mathbb Z}[\frac{1}{p}].\) This generalization of the Chow weight structure from the case of \({\mathbb Q}\)-linear motives is achieved by generalizing techniques (``gluing construction'') from [\textit{M. V. Bondarko}, J. K-Theory 6, No. 3, 387--504 (2010; Zbl 1303.18019); Int. Math. Res. Not. 2014, No. 17, 4715--4767 (2014; Zbl 1400.14062)].