Equivalent adic filtrations have canonically isomorphic completions

For each $k\ge 1$, the inclusion $I^c\subseteq J$ gives $I^{ck}M\subseteq J^{k}M$ and hence a quotient map $M\mathrm{/}{I}^{ck}M\to M\mathrm{/}{J}^{k}M\mathrm{.}$ The powers $ck$ form a cofinal subsystem of the $I$-adic tower. The displayed maps are compatible, so [L2] induces an $R$-linear map $\mathrm{\Phi }:{\widehat{M}}^I\to {\widehat{M}}^J\mathrm{.}$

Similarly, $J^d\subseteq I$ gives compatible quotient maps $M\mathrm{/}{J}^{dk}M\to M\mathrm{/}{I}^{k}M$ on a cofinal subsystem and hence a map $\mathrm{\Psi }:{\widehat{M}}^J\to {\widehat{M}}^I\mathrm{.}$

On either side, composing the two systems of quotient maps eventually reduces modulo a larger and larger power of the same ideal. Hence the composites $\mathrm{\Psi }\mathrm{\Phi }$ and $\mathrm{\Phi }\mathrm{\Psi }$ induce the identity on every finite stage, so they are the identity on the inverse limits. Therefore $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are inverse isomorphisms.

Thus equivalent adic filtrations have canonically isomorphic completions.