(Formal scheme)
Let \(A\) be a noetherian ring, complete and separated in the \(I\)-adic topology for an ideal \(I\). The formal spectrum \(\operatorname{Spf} A\) is the topological space \(\Spec(A/I)\) with the sheaf of topological rings \(\varprojlim_{n\ge1}\OO_{\Spec(A/I^n)}\). A locally noetherian formal scheme is a topologically ringed space locally isomorphic to such formal spectra. It is a noetherian formal scheme if it is also quasi-compact, equivalently if finitely many of these affine formal opens cover it [@Har10a, Chapter II, §9]. Its underlying space is then noetherian: each affine formal open has the topology of a noetherian affine scheme, and a finite union of noetherian open subspaces is noetherian.
(Completion along a closed subscheme)
Let \(X\) be a Noetherian scheme and \(Z \subseteq X\) a closed subscheme with ideal sheaf \(\mathcal{I}\). The formal completion of \(X\) along \(Z\) is \(\hat{X} = (Z, \varprojlim_n \OO_X/\mathcal{I}^n)\), a Noetherian formal scheme with underlying space \(Z\). For \(X = \Spec A\) and \(Z = V(I)\), \(\hat{X} = \operatorname{Spf} \hat{A}\) with \(\hat{A} = \varprojlim_n A/I^n\).
(The formal disc)
\(\Spec k[[t]]\) has two points: the closed point \((t)\) and the generic point \((0)\), whose residue field is \(k((t))\). It plays the role of a small disc \(\mathbb{D}\) around \(0\) in the \(t\)-line, and its open subscheme \(\Spec k((t))\), the generic point, plays the role of the punctured disc \(\mathbb{D} \setminus \{0\}\). The formal completion of \(\AA^1_k\) at the origin is \(\operatorname{Spf} k[[t]]\), whose underlying space is the single point \(0\) and which remembers every infinitesimal neighbourhood \(\Spec k[t]/(t^n)\).
Dependencies
D-VKR54$\Spec A$ as a locally ringed spaceuses