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ON MORPHISMS KILLING WEIGHTS AND STABLE HUREWICZ-TYPE THEOREMS
Published online by Cambridge University Press: 24 October 2022
Abstract
For a weight structure w on a triangulated category $\underline {C}$ we prove that the corresponding weight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerate objects’; this improves earlier conservativity formulations. In the case
$w=w^{sph}$ (the spherical weight structure on
$SH$), we deduce the following converse to the stable Hurewicz theorem:
$H^{sing}_{i}(M)=\{0\}$ for all
$i<0$ if and only if
$M\in SH$ is an extension of a connective spectrum by an acyclic one. We also prove an equivariant version of this statement.
The main idea is to study M that has no weights $m,\dots ,n$ (‘in the middle’). For
$w=w^{sph}$, this is the case if there exists a distinguished triangle
$LM\to M\to RM$, where
$RM$ is an n-connected spectrum and
$LM$ is an
$m-1$-skeleton (of M) in the sense of Margolis’s definition; this happens whenever
$H^{sing}_i(M)=\{0\}$ for
$m\le i\le n$ and
$H^{sing}_{m-1}(M)$ is a free abelian group. We also consider morphisms that kill weights
$m,\dots ,n$; those ‘send n-w-skeleta into
$m-1$-w-skeleta’.
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 2 , March 2024 , pp. 521 - 556
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press
References
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