2024/11/24 by Matteo Casarosa, Chris Lambie-Hanson, Chris Lambie‐Hanson +2 · 1 citation
Decision Sciences · #03E05 #03E17 #03E35 #03E75 #18G10 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2411.15856
The derived functors limn of the inverse limit find many applications in algebra and topology. In particular, the vanishing of certain derived limits limn A[H], parametrized by an abelian group H, has implications for strong homology and condensed mathematics. In this paper, we prove that if \mathfrakd=ωn, then limn A[H] ≠ 0 holds for H=ℤ(ωn) (i.e. the direct sum of ωn-many copies of ℤ). The same holds for H=ℤ under the assumption that w\diamondsuit(Sk+1k) holds for all k < n. In particular, this shows that if limn A[H] = 0 holds for all n ≥ 1 and all abelian groups H, then 2ℵ0 ≥ ℵω+1, thus answering a question of Bannister. Finally, we prove some consistency results regarding simultaneous nonvanishing of derived limits, again in the case of H = ℤ. In particular, we show the consistency, relative to ZFC, of \bigwedge2 ≤ k < ω limk A ≠ 0.