Gersten weight structures for motivic homotopy categories; retracts of cohomology of function fields, motivic dimensions, and coniveau spectral sequences

TL;DR

This paper develops Gersten weight structures in motivic homotopy categories to analyze cohomology theories, proving functoriality of coniveau spectral sequences and splitting properties of Cousin complexes for schemes.

Contribution

Introduction of Gersten weight structures for motivic pro-spectra categories, enabling new results on cohomology functoriality and splitting in motivic homotopy theory.

Findings

Functoriality of coniveau spectral sequences for cohomology theories

Splitting of Cousin complexes for semi-local schemes

Cohomology of schemes as direct summands of open dense subschemes

Abstract

For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $SH^{S^1}(k)$ we establish the $SH^{S^1}(k)$-functoriality of coniveau spectral sequences for $H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin complex for $H^*(S)$ splits; if $H$ also factorizes through $SH^+(k)$ or $DM(k)$, then this is also true for any primitive $S$. Moreover, the cohomology of such an $S$ is a direct summand of the cohomology of any its open dense subscheme. This is a vast generalization of the results of a previous paper. In order to prove these results we consider certain triangulated categories of motivic pro-spectra, and introduce Gersten weight structures for them. We study in detail the notions of cohomological dimensions of scheme associated to various categories of motivic pro-spectra; they are defined in terms of the corresponding weight structures and count the number of non-zero Nisnevich cohomology for sheaves in the hearts of orthogonal "homotopy" $t$-structures.