Killing weights from the perspective of t-structures

TL;DR

This paper investigates morphisms that eliminate specific weights in triangulated categories with weight structures, providing new criteria, decompositions, and characterizations relevant to homotopy theory and cohomology.

Contribution

It introduces novel criteria for morphisms and objects to kill weights, utilizing virtual t-truncations and adjacent t-structures, and constructs new torsion theories and decompositions.

Findings

Morphisms killing weights can be characterized by factorization through objects without these weights.

New criteria involving virtual t-truncations and adjacent t-structures are established.

Weakly functorial decompositions of spectra and descriptions of morphisms acting trivially on cohomology are obtained.

Abstract

This 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 satisfy these conditions. 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 degree zero singular cohomology with coefficients in every abelian group.