Killing Weights from the Perspective of $t$-Structures

Mikhail V. Bondarko ; Sergei V. Vostokov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 59-70

Résumé

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 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 $m,\dots ,n$ 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 $\mathrm {SH}$) and a new description of those morphisms that act trivially on the singular cohomology $H_{\mathrm{sing}}^0(-,\Gamma )$ with coefficients in an arbitrary abelian group $\Gamma $.

Détail

Keywords: triangulated category, weight structure, killing weights, objects without weights, torsion theory, projective class, injective class, stable homotopy category, singular (co)homology.
Mots-clés : $t$-structure