On Morphisms Killing Weights and Stable Hurewicz-Type Theorems

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’.

On perfectly generated weight structures and adjacent t-structures

“…Firstly, conditions 1 and 2 are equivalent by Corollary 3.1.4 (resp. Theorem 3.1.3) of [Bon22b]. Moreover, these statements also yield that condition 2 follows from 3; recall here that pure functors (cf.…”

“…Applying the uniqueness statement in Proposition 1.2.4(4) one can easily obtain T 1 ∼ = T 2 . In particular, one can note that both T 1 and T 2 give weight decompositions of M that avoid weight n 2 − 1; see Theorem 2.2.1(9) of [Bon22b]. Now we applying loc.…”

“…Consequently, the uniqueness provided by Proposition 1.2.4(4) yields condition 7; cf. Remark 1.2.6 of [Bon22b] where compositions of the corresponding morphisms between decomposition triangles is discussed.…”

“…Remark 4.1.5. If w is weight-Karoubian, that is, if Hw is Karoubian then one modify the definition of essentially w-bounded above and below objects by setting M = M in the triangles (4.1.1); see Theorems 2.3.4 and 3.1.3 and Corollary 3.1.4 of[Bon22b]. Respectively, one can compute τ ≤m (H)(M ) for any M ∈ Obj C using the corresponding canonical decomposition similarly to Proposition 4.1.4(1).…”