The Stacks project

Example 15.93.4. Let $p$ be a prime number. Consider the map $\mathbf{Z}_ p[x] \to \mathbf{Z}_ p[y]$ of polynomial algebras sending $x$ to $py$. Consider the cokernel $M = \mathop{\mathrm{Coker}}(\mathbf{Z}_ p[x]^\wedge \to \mathbf{Z}_ p[y]^\wedge )$ of the induced map on (ordinary) $p$-adic completions. Then $M$ is a derived complete $\mathbf{Z}_ p$-module by Proposition 15.91.5 and Lemma 15.91.6; see also discussion in Example 15.93.3. However, $M$ is not $p$-adically complete as $1 + py + p^2 y^2 + \ldots $ maps to a nonzero element of $M$ which is contained in $\bigcap p^ nM$.