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Let $B$ be a (torsion free) $\mathbb{Z}$-module. Consider the following tower: $$T=\dots\to B\to B\to B$$ where each map is the multiplication by $p\in\mathbb{Z}$. What is $\underset{\mathrm{\leftarrow}}{lim}^1 T$? Using $p$-adic expansion, I can show that it is $\mathbb{Z}_p/\mathbb{Z}$ when $B=\mathbb{Z}$, but I cannot deal with the general case.

Gregg
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1 Answers1

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You can compute $\varprojlim^1$ of such a tower $T =\{f_n : A_{n+1}\to A_n\}$ as a cokernel. Concretely, consider the map $$F: \prod_{n\geqslant 0} A_n\to \prod_{n\geqslant 0} A_n$$ which has components $A_n\times A_{n+1} \to A_n$ given by $(x,x') \mapsto x-f_n(x')$. Then $\varprojlim^1 T = \operatorname{coker} F$. This should then allow you to compute $\varprojlim^1 T$ in you case.

Pedro
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  • Thank you for answer. I understand this, you may just assume that this is my definition of $\mathrm{lim}^1$. But the problem is that I cannot explicitly describe this cokernel (sorry, now I think that I should have said this explicitly in my question). Namely, I think that for torsion free B, it should be just $\hat{B}_p/B$, but I cannot see why. – Gregg Oct 10 '18 at 09:50
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    @Gregg Ah, I was somewhat suspecting this. Did you try to see if your proof of the $B=\mathbb Z$ case simply extends to the torsion-free case? – Pedro Oct 10 '18 at 10:32
  • Sorry for such a late response. Yes, indeed, my proof works absolutely fine in this case. The key observation is that for torsion free $B$, the p-adic expansion of any $b\in\hat{B}$ turns out to be unique. – Gregg Oct 22 '18 at 21:03