ON MORPHISMS KILLING WEIGHTS AND STABLE HUREWICZ-TYPE THEOREMS

Abstract

For a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case $w=w^{sph}$ (the spherical weight structure on $SH$ ), we deduce the following converse to the stable Hurewicz theorem: $H^{sing}_{i}(M)=\{0\}$ for all $i<0$ if and only if $M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement. The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For $w=w^{sph}$ , this is the case if there exists a distinguished triangle $LM\to M\to RM$ , where $RM$ is an n-connected spectrum and $LM$ is an $m-1$ -skeleton (of M) in the sense of Margolis’s definition; this happens whenever $H^{sing}_i(M)=\{0\}$ for $m\le i\le n$ and $H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights $m,\dots ,n$ ; those ‘send n-w-skeleta into $m-1$ -w-skeleta’.