Lemma 87.5.3. Let $A$ be a linearly topologized ring. The map $A \to A^\wedge $ from $A$ to its completion is taut.

**Proof.**
Let $I_\lambda $ be a fundamental system of open ideals of $A$. Recall that $A^\wedge = \mathop{\mathrm{lim}}\nolimits A/I_\lambda $ with the limit topology, which means that the kernels $J_\lambda = \mathop{\mathrm{Ker}}(A^\wedge \to A/I_\lambda )$ form a fundamental system of open ideals of $A^\wedge $. Since $J_\lambda $ is the closure of $I_\lambda A^\wedge $ (compare with Lemma 87.4.11) we conclude.
$\square$

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