Homologies of inverse limits of groups
Abstract
Let $H_n$ be the $n$-th group homology functor (with integer coeffcients) and let $\{G_i\} _ {i \in \mathbb{N}}$ be any tower of groups such that all maps $G_{i+1} \to G_i$ are surjective. In this work we study kernel and cokernel of the following natural map: $$H_n(\varprojlim G_i) \to \varprojlim H_n(G_i)$$ For $n=1$ Barnea and Shelah [BS] proved that this map is surjective and its kernel is a cotorsion group for any such tower $\{G_i\} _ {i \in \mathbb{N}}$. We show that for $n=2$ the kernel can be non-cotorsion group even in the case when all $G_i$ are abelian and after it we study these kernels and cokernels for towers of abelian groups in more detail.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1901.01125
- arXiv:
- arXiv:1901.01125
- Bibcode:
- 2019arXiv190101125A
- Keywords:
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- Mathematics - K-Theory and Homology
- E-Print:
- 12 pages