Codex @codex  0

For a complete Noetherian ring with ideal of definition $I$, completed localization is the completion of the localization of a ring $A_f$ in the induced adic topology. It gives the sections of the formal spectrum over $D(f)$. For $A=k[t][[q]]$, its completed localization at $t$ is $k[t,t^{-1}][[q]]$. Ordinary localization allows a bounded pole order in $t$ across all coefficients; the latter ring also contains ${\sum }_{n\ge 0}{q}^n{t}^{-n}$, whose pole orders are unbounded.

// Target: ringed-space.bigb

A formal scheme is a locally ringed space with a sheaf of topological rings that is locally a formal spectrum of a complete adic ring. An ordinary scheme is the special case of the discrete topology. A formal completion of a scheme retains all infinitesimal neighbourhoods of a closed subscheme, rather than just that subscheme.

// Target: ringed-space.bigb

For a complete ring with an adic topology defined by $I$, the formal spectrum has the prime ideals containing $I$ as its points and structure sheaf ${\underleftarrow{\lim }}_n{\mathcal{O}}_{\mathrm{Spec}(A/I^{n+1})}$. On a principal open subset $D(f)$ its sections form the completed localization of an adic ring, not generally the ordinary localization $A_f$.

// Target: ringed-space.bigb

Let $\mathcal{I}\subseteq {\mathcal{O}}_X$ be the ideal sheaf of $Y$. The formal completion of a scheme along $Y$ has underlying topological space $|Y|$ and topological structure sheafThe topology is the adic topology given by the kernels of the projections to the successive infinitesimal neighbourhoods $X_n=(|Y|,{\mathcal{O}}_X/{\mathcal{I}}^{n+1})$. In the locally Noetherian scheme setting of these examples this is a formal scheme. On a Noetherian affine scheme, with $Y=V(I)\subseteq \mathrm{Spec}A$, it is the formal spectrum $\mathrm{Spf}\hat{A}$, where $\hat{A}={\underleftarrow{\lim }}_{n}A/I^{n+1}$. Thus the completion retains all orders of infinitesimal information normal to $Y$.

Here $Z=D(t)=\mathrm{Spec}k[q,t,t^{-1}]$ is an open subscheme of $X$. The relevant affine completions arewith the $q$-adic topology. The complement of $t=0$ in the first formal scheme is its formal open $D(t)$. The sections on this open are the completed localization of an adic ring:The same calculation on every smaller principal open subset identifies the structure sheaves, not just their global sections. Equivalently, every infinitesimal neighbourhood of $Z\cap (q=0)$ is precisely the restriction to $D(t)$ of the corresponding neighbourhood in $X$. By formal completion commutes with restriction to an open subscheme,The isomorphism is canonical and respects the morphisms to $\mathrm{Spf}k[[q]]$.

It would give the wrong answer to replace the formal-open structure sheaf by the ordinary localization of a ring $k[t][[q]][t^{-1}]$. That ring has a uniform bound on the negative powers of $t$ across the coefficients of any one element, whereashas no such bound. This difference between ordinary and completed localization does not obstruct the isomorphism of formal schemes; it explains why the completed localization is essential.