TY - JOUR

T1 - On morphisms killing weights and stable Hurewicz-type theorems

AU - Бондарко, Михаил Владимирович

PY - 2024

Y1 - 2024

N2 - For a weight structure w on a triangulated category C we prove that the correspondingweight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerateobjects’; this improves earlier conservativity formulations. In the case w = w sph (the spherical weightstructure on SH), we deduce the following converse to the stable Hurewicz theorem: H singi(M) = {0}for all i < 0 if and only if M ∈ SH is an extension of a connective spectrum by an acyclic one. We alsoprove an equivariant version of this statement.The main idea is to study M that has no weights m,...,n (‘in the middle’). For w =w sph , this is the caseif there exists a distinguished triangle LM → M → RM, where RM is an n-connected spectrum and LMis an m−1-skeleton (of M) in the sense of Margolis’s definition; this happens whenever H singi(M) = {0}for m ≤ i ≤ n and H singm−1 (M) is a free abelian group. We also consider morphisms that kill weightsm,...,n; those ‘send n-w-skeleta into m−1-w-skeleta’.

AB - For a weight structure w on a triangulated category C we prove that the correspondingweight complex functor and some other (weight-exact) functors are ‘conservative up to weight-degenerateobjects’; this improves earlier conservativity formulations. In the case w = w sph (the spherical weightstructure on SH), we deduce the following converse to the stable Hurewicz theorem: H singi(M) = {0}for all i < 0 if and only if M ∈ SH is an extension of a connective spectrum by an acyclic one. We alsoprove an equivariant version of this statement.The main idea is to study M that has no weights m,...,n (‘in the middle’). For w =w sph , this is the caseif there exists a distinguished triangle LM → M → RM, where RM is an n-connected spectrum and LMis an m−1-skeleton (of M) in the sense of Margolis’s definition; this happens whenever H singi(M) = {0}for m ≤ i ≤ n and H singm−1 (M) is a free abelian group. We also consider morphisms that kill weightsm,...,n; those ‘send n-w-skeleta into m−1-w-skeleta’.

U2 - 10.1017/S1474748022000470

DO - 10.1017/S1474748022000470

M3 - Article

VL - 23

SP - 521

EP - 556

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 2

ER -