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 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 \(H_{\text{sing}}^0(-,\Gamma)\) with coefficients in an arbitrary abelian group \(\Gamma\).
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Notes
Recall also that D. Pauksztello has introduced weight structures independently (in [13]); he called them co-\(t\)-structures.
In earlier texts of the authors, connective subcategories were called negative ones; another related notion is silting.
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Acknowledgments
The authors are deeply grateful to the referee for useful comments. The first author also thanks the Max Planck Institut für Mathematik for their hospitality during the completion of this version of the paper.
Funding
This work was supported by the Russian Science Foundation under grant no. 20-41-04401, https://rscf.ru/project/20-41-04401/.
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Vol. 320, pp. 59–70 https://doi.org/10.4213/tm4299.
Dedicated to Alexey Nikolaevich Parshin, whose memories will live on long after he has passed
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Bondarko, M.V., Vostokov, S.V. Killing Weights from the Perspective of \(t\)-Structures. Proc. Steklov Inst. Math. 320, 51–61 (2023). https://doi.org/10.1134/S0081543823010042
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DOI: https://doi.org/10.1134/S0081543823010042