On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra

Abstract

We ask whether a morphism $g$ in a triangulated category $C$ endowed with a weight structure "kills weights" (between an integer $m$ and some $n\ge m$). If $g=id_M$ (where $M\in Obj C$) and $C$ is Karoubian, then $g$ kills weights $m,\dots,n$ whenever there exists a weight decomposition of $M$ that "avoids" these weights (in the sense earlier defined by Wildeshaus). We prove the equivalence of several definitions for killing weights. In particular, we describe a family of cohomological functors that "detects" this notion. We also prove that $M$ is without weights $m,\dots, n$ (i.e., a decomposition of $M$ avoiding these weights exists) if and only if the corresponding condition is fulfilled for its weight complex $t(M)$. These results allow us to get new (stronger) results on the conservativity of the weight complex functor $t$. We study in detail the case $C=SH$ (endowed with the spherical weight structure whose heart consists of coproducts of sphere spectra); the corresponding weight complex functor is just the one calculating the $H\mathbb{Z}$-homology (whose terms are free abelian groups). In this case $g$ kills weights $m,\dots, n$ if and only if $H(g)=0$ for all $H$ represented by elements of $SH[m,n]$ (so, these morphisms form an injective class of morphisms in the sense defined by Christensen; yet this class is not stable with respect to shifts). Moreover, for any spectrum $M$ there exists a "weakly universal decomposition" $P\to M\to I_0$ for $I_0\in SH[m,n]$ and $P$ being without weights $m,\dots,n$ (so, we obtain a torsion pair). We also prove a certain converse to the stable Hurewicz theorem.