Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences

Abstract

For any cohomology theory $H$ that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) $S{H}^{S^1}(k)$ (or through $SH(k)$) we establish the $S{H}^{S^1}(k)$-functorialty (resp. $SH(k)$-one) of coniveau spectral sequences for $H$. We also prove: for any affine essentially smooth semi-local $S$ the Cousin complex for $H^{\ast }(S)$ splits; if $H$ also factorizes through $S{H}^{+}(k)$ or $S{H}^{MGL}(k)$, 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 categories of motivic pro-spectra, and introduce Gersten weight structures for them. Our results rely on several interesting statements on weight structures in cocompactly cogenerated triangulated categories and on the '$S{H}^{+}(k)$-acyclity' of primitive schemes. .

Several minor corrections made; this includes the localization by a set of primes issue (see Remarks 2.2.7 and 4.4.2)

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