The Stacks project

Example 15.91.4. 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 15.90.5 and Lemma 15.90.6; see also discussion in Example 15.91.3. 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 $\bigcap p^ nM$.

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