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

2013/12/29 by Bondarko, Mikhail V.
#18E30 #18G40 #18G55 #19D25 #19E15 #55P42. Secondary: 14C15 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary: 14F42

paper · doi:10.48550/arxiv.1312.7493

Abstract

For any cohomology theory H that can be factorized through (the Morel-Voevodsky's triangulated motivic homotopy category) SH^S1(k) (or through SH(k)) we establish the SH^S1(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^*(S) splits; if H also factorizes through SH⁺(k) or SH^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 'SH⁺(k)-acyclity' of primitive schemes. .

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