Kernel and universal property of adic completion

The completion map ${\kappa }_M$ is the unique map into the inverse limit of [L1] whose $n$th component, for $n\ge 1$, is the quotient map $q_n\text{\hspace{0.17em}⁣}:M\to M\mathrm{/}{I}^{n}M\mathrm{.}$

An element lies in $\ker ({\kappa }_M)$ exactly when every quotient map $q_n$, $n\ge 1$, kills it, equivalently when it belongs to every $I^{n}M$ for $n\ge 1$. Since $I^{0}M=M$, this gives $\ker ({\kappa }_M)={\bigcap }_{n\ge 1}{I}^{n}M={\bigcap }_{n\ge 0}{I}^{n}M\mathrm{.}$

Let $f\text{\hspace{0.17em}⁣}:M\to N$ be $R$-linear with $N$ complete. Since $f$ is $R$-linear, one has $f(I^{n}M)\subseteq I^{n}N$ for every $n\ge 1$, so $f$ induces maps $f_n\text{\hspace{0.17em}⁣}:M\mathrm{/}{I}^{n}M\to N\mathrm{/}{I}^{n}N\mathrm{.}$ These maps are compatible with the quotient transition maps.

By [L4], the compatible family $(f_n)$ yields a unique map such that the $n$th projection of $\tilde{f}{\kappa }_M$ is $f_n{q}_n$. These are also the coordinates of ${\kappa }_{N}f$, so uniqueness in [L4] gives $\tilde{f}{\kappa }_M={\kappa }_{N}f\mathrm{.}$

The image of ${\kappa }_M$ is dense for the inverse-limit topology. Indeed, for $x\in \widehat{M}$ and $n\ge 1$, choose $m\in M$ representing ${\pi }_n^M(x)\in M\mathrm{/}{I}^{n}M$; then $x-{\kappa }_M(m)\in K_n$. Now let $g:\widehat{M}\to N$ be continuous and vanish on ${\kappa }_M(M)$. If $g(x)\mathrm{\ne }0$, separatedness of $N$ from [L2] gives $n$ with $g(x)\mathrm{\notin }{I}^{n}N$. The inverse image $g^{-1}(g(x)+I^{n}N)$ is an open neighbourhood of $x$ by [L3], so density gives some ${\kappa }_M(m)$ in it. But $g{\kappa }_M(m)=0$, which would put $g(x)$ in $I^{n}N$, a contradiction. Hence $g=0$. Applying this to the difference of two continuous extensions proves uniqueness.

The last equality shows that ${\kappa }_M$ is injective exactly when ${\bigcap }_{n\ge 0}{I}^{n}M=0$, which is exactly the definition of being $I$-adically separated.

Write ${\pi }_n^M:\widehat{M}\to M\mathrm{/}{I}^{n}M$ and . Step 1.4 gives ${\pi }_n^N\tilde{f}=f_n{\pi }_n^M,$ so $\tilde{f}(K_n)\subseteq \ker ({\pi }_n^N)$ for every $n\ge 1$. Hence $\tilde{f}$ is continuous for the two inverse-limit topologies. Because $N$ is complete, [L2] identifies under this isomorphism the preimage of $\ker ({\pi }_n^N)$ is $I^{n}N$, so [L3] makes ${\kappa }_N$ a homeomorphism from the $I$-adic topology to the inverse-limit topology. Thus $\widehat{f}:={\kappa }_N^{-1}\tilde{f}:\widehat{M}\to N$ is continuous and satisfies $\widehat{f}{\kappa }_M=f$.

Parts 1 and 2 are proved by steps 1.2, 1.5, 2.1, and 2.2.