Lim^1 of a tower

Question

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.

Answer 1

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.