  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.
