Killing Weights from the Perspective of t-Structures

paper is devoted to morphisms killing weights in a range (as defined by the first author) and to objects without these weights (as essentially defined by J. Wildeshaus) in a triangulated category endowed with a weight structure w. We describe several new criteria for morphisms and objects to be of these types. In some of them we use virtual t-truncations and a t-structure adjacent to w. In the case where the latter exists, we prove that a morphism kills weights m, ... , n if and only if it factors through an object without these weights; we also construct new families of torsion theories and projective and injective classes. As a consequence, we obtain some "weakly functorial decompositions" of spectra (in the stable homotopy category SH) and a new description of those morphisms that act trivially on the singular cohomology H-sing(0)(-, G) with coefficients in an arbitrary abelian group G.