1 Analytic spectrum of an algebra of finite presentation
Let S be a ringed space. We say that an {\mathscr{O}}_S-algebra {\mathscr{A}} is of finite presentation if every s\in S admits a neighbourhood U such that {\mathscr{A}}|U is isomorphic to a sheaf of the form \frac{({\mathscr{O}}_S|U)[t_1,\ldots,t_n]}{(f_1,\ldots,f_m)} with f_1,\ldots,f_m\in\Gamma(U,{\mathscr{O}}_S[t_1,\ldots,t_n]).
That is, {\mathscr{A}} is locally generated over {\mathscr{O}}_S by a finite number of sections subject to a finite number of relations.
We are interested in the case where S is an analytic space, over a non-discrete complete valued base field k. For every analytic space T over S, consider the sheaf {\mathscr{A}}_T={\mathscr{A}}\times_S T, the inverse image of {\mathscr{A}} on T, and the set F_{\mathscr{A}}(T)=\operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{A}}_T,{\mathscr{O}}_T) of algebra homomorphisms; we immediately see that T\mapsto F_{\mathscr{A}}(T) defines a contravariant functor F_{\mathscr{A}}\colon(\mathsf{An})^\mathrm{op}/S\to\mathsf{Set}.
Let {\mathscr{A}} be an algebra of finite presentation on an analytic space S; the functor F_{\mathscr{A}} is representable in the category \mathsf{An}/S by an analytic space separated over S.
Proof. We note first of all that the functor F_{\mathscr{A}} is of a local nature (cf. Exposé 11, Definition 5.4), since if T is an analytic space over S, then the presheaf U\mapsto F_{\mathscr{A}}(U) (where U runs over the opens of T) is precisely the sheaf of germs of algebra homomorphisms \underline{\operatorname{Hom}}_{{\mathscr{O}}_T}({\mathscr{A}}_T,{\mathscr{O}}_T).
Thus, by Corollary 5.7 of Exposé 11, it suffices, to show that F_{\mathscr{A}} is representable, to find an open cover (S_i) of S such that, for each index i, the functor F_{\mathscr{A}}/S_i is representable in \mathsf{An}/S_i.
By Definition 1, we thus see that it suffices to consider the case where {\mathscr{A}} is of the form {\mathscr{O}}_S[t_1,\ldots,t_n]/(f_1,\ldots,f_m). Then the exact sequence 0 \to {\mathscr{I}}\xrightarrow{i} {\mathscr{O}}_S[t_1,\ldots,t_n] \to {\mathscr{A}}\to 0 (where {\mathscr{I}}=(f_1,\ldots,f_m)) gives, by base extension, an exact sequence {\mathscr{I}}_T \xrightarrow{i_T} {\mathscr{O}}_T[t_1,\ldots,t_n] \to {\mathscr{A}}_T \to 0 for every analytic space T over S. This allows us to identify F_{\mathscr{A}}(T)=\operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{A}}_T,{\mathscr{O}}_T) with the set of homomorphisms from {\mathscr{O}}_T[t_1,\ldots,t_n] to {\mathscr{O}}_T that vanish on i_T({\mathscr{I}}_T).
But the functor F_{{\mathscr{O}}_S[t_1,\ldots,t_n]}\colon T\mapsto\operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{O}}_T[t_1,\ldots,t_n],{\mathscr{O}}_T) is isomorphic to T\mapsto(\Gamma(T,{\mathscr{O}}_T))^n; by Theorem 1.1 of Exposé 10 and Proposition 3.1 of Exposé 11, we see that F_{{\mathscr{O}}_S[t_1,\ldots,t_n]} is represented in \mathsf{An}/S by the pair (S\times{\mathscr{E}}^n,\eta), where \eta is the homomorphism of algebras \eta\colon{\mathscr{O}}_{S\times{\mathscr{E}}^n}[t_1,\ldots,t_n]\to{\mathscr{O}}_{S\times{\mathscr{E}}^n} defined by \eta(t_i)=q^*(z_i) (for i=1,\ldots,n) and denoting by q the projection from S\times{\mathscr{E}}^n onto {\mathscr{E}}^n. It immediately follows that the sub-functor F_{\mathscr{A}} is represented by the closed subspace X\subset S\times{\mathscr{E}}^n defined by the ideal {\mathscr{I}}=\eta\circ i_{S\times{\mathscr{E}}^n}({\mathscr{I}}_{S\times{\mathscr{E}}^n})\subset{\mathscr{O}}_{S\times{\mathscr{E}}^n} (cf. Exposé 11, Lemma 3.6).
In the case in question, X is evidently separated over S. Since the property of being separated over S, for an object of \mathsf{An}/S, is local on S, the latter claim of the statement is also proven (cf. Expose 11, Proposition 5.6: we obtain the space that represents F_{\mathscr{A}} by gluing together those that represent the F_{{\mathscr{A}}/S_i}).
For every {\mathscr{O}}_S-algebra {\mathscr{A}} of finite presentation, we define the analytic spectrum of {\mathscr{A}}, denoted \operatorname{Specan}({\mathscr{A}}), to be the object of \mathsf{An}/S (defined up to S-isomorphism) that represents the functor F_{\mathscr{A}}\colon T\longmapsto \operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{A}}_T,{\mathscr{O}}_T). \operatorname{Specan}({\mathscr{A}}) is thus an analytic space separated over S.
Consider an {\mathscr{O}}_S-module {\mathscr{E}} of finite presentation; its symmetric algebra S({\mathscr{E}}) is an {\mathscr{O}}_S-algebra of finite presentation, and \operatorname{Specan}(S({\mathscr{E}})) represents the functor T\longmapsto \operatorname{Hom}_{{{\mathscr{O}}_T}\text{-}\mathsf{Alg}}(S({\mathscr{E}}_T),{\mathscr{O}}_T) \simeq \operatorname{Hom}_{{{\mathscr{O}}_T}\text{-}\mathsf{Mod}}({\mathscr{E}}_T,{\mathscr{O}}_T). This is the vector bundle defined by {\mathscr{E}} (cf. Exposé 12, §1).
- {\mathscr{A}}\mapsto\operatorname{Specan}({\mathscr{A}}) defines a contravariant functor from the category of {\mathscr{O}}_S-algebras of finite presentation to \mathsf{An}/S.
- For any {\mathscr{O}}_S-algebras {\mathscr{A}} and {\mathscr{B}} of finite presentation, {\mathscr{A}}\otimes_{{\mathscr{O}}_S}{\mathscr{B}} is of finite presentation, and \operatorname{Specan}({\mathscr{A}}\otimes_{{\mathscr{O}}_S}{\mathscr{B}})\simeq\operatorname{Specan}({\mathscr{A}})\times_S\operatorname{Specan}({\mathscr{B}}).
- Let h\colon{\mathscr{A}}\to{\mathscr{B}} be a surjective homomorphism of algebras of finite presentation. Then the corresponding morphism \operatorname{Specan}(h)\colon\operatorname{Specan}({\mathscr{B}})\to\operatorname{Specan}({\mathscr{A}}) is a closed immersion.
- For every {\mathscr{O}}_S-algebra {\mathscr{A}} of finite presentation, and for every analytic space T over S, \operatorname{Specan}({\mathscr{A}})\times_S T\simeq\operatorname{Specan}({\mathscr{A}}_T) (compatibility with base change).
Proof.
- is immediate, since F_{\mathscr{A}}(T) is a contravariant functor in {\mathscr{A}};
- follows from the formula \operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{A}}_T\otimes_{{\mathscr{O}}_T}{\mathscr{B}}_T,{\mathscr{O}}_T) \simeq \operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{A}}_T,{\mathscr{O}}_T)\times\operatorname{Hom}_{{\mathscr{O}}_T}({\mathscr{B}}_T,{\mathscr{O}}_T)
- follows easily from the proof of Proposition 1, and (iv) can be proven by Corollary 3.2 of Exposé 11.
Consider the pair (X,\xi) that represents the functor F_{\mathscr{A}} defined by an {\mathscr{O}}_S-algebra {\mathscr{A}} of finite presentation. Then X=\operatorname{Specan}({\mathscr{A}}) is an analytic space over S, with structure morphism f\colon X\to S, and \xi\in F_{\mathscr{A}}(X)=\operatorname{Hom}_{{\mathscr{O}}_X}({\mathscr{A}}_X,{\mathscr{O}}_X) is a homomorphism from {\mathscr{A}}_X=f^*({\mathscr{A}}) to {\mathscr{O}}_X. In other words, \xi defines a factorisation X \xrightarrow{\hat{f}} (|S|,{\mathscr{A}}) \to S of the structure morphism f of X through the ringed space (|S|,{\mathscr{A}}) (where |S| denotes the underlying topological space of S) endowed with the canonical morphism (|S|,{\mathscr{A}})\to S (consisting of the identity map on |S| and the homomorphism {\mathscr{O}}_S\to{\mathscr{A}}).
For any {\mathscr{A}}-module {\mathscr{M}}, set \widetilde{{\mathscr{M}}} = \hat{f}^*({\mathscr{M}}) = f^*({\mathscr{M}})\otimes_{f^*({\mathscr{A}})}{\mathscr{O}}_X. We thus define a functor {\mathscr{M}}\mapsto\widetilde{{\mathscr{M}}} from the category of {\mathscr{A}}-modules to the category of {\mathscr{O}}_X-modules (where X=\operatorname{Specan}({\mathscr{A}})). If {\mathscr{M}} and {\mathscr{N}} are {\mathscr{A}}-modules then ({\mathscr{M}}\otimes_{\mathscr{A}}{\mathscr{N}})^\sim = \widetilde{{\mathscr{M}}}\otimes_{{\mathscr{O}}_X}\widetilde{{\mathscr{N}}}.
Consider an analytic space T over S, and the space X_T = \operatorname{Specan}({\mathscr{A}}_T) = X\times_S T over T ((iv) of Proposition 2); the structure morphism from X_T to T again factors as X_T \xrightarrow{\hat{f}_T} (|T|,{\mathscr{A}}_T) \to T and we obtain a commutative diagram \begin{CD} X @<<< X_T \\@V{\hat{f}}VV @VV{\hat{f}_T}V \\(|S|,{\mathscr{A}}) @<<< (|T|,{\mathscr{A}}_T) \end{CD} Let {\mathscr{M}} be an {\mathscr{A}}-module; the inverse image {\mathscr{M}}_T={\mathscr{M}}\times_S T of {\mathscr{M}} over T is an {\mathscr{A}}_T-module. We thus see that \widetilde{{\mathscr{M}}}_T=\hat{f}_T^*({\mathscr{M}}_T) can be identified with the inverse image of \widetilde{{\mathscr{M}}} under the morphism X_T\to X (compatibility of the functor {\mathscr{M}}\mapsto\widetilde{{\mathscr{M}}} with base change).
2 Fibre of the analytic spectrum over a point in the base
Consider an {\mathscr{O}}_S-algebra {\mathscr{A}} of finite presentation, and its analytic spectrum X=\operatorname{Specan}({\mathscr{A}}). If s\in S, recall (Exposé 10, §2) that the fibre of X at s is the fibre product X_s=X\times_S\{s\}, where \{s\} is the analytic space consisting of the point s, endowed with the field {\mathscr{O}}_s/{\mathfrak{m}}_s{\mathscr{O}}_s=K(s)\simeq k (where {\mathfrak{m}}_s denotes the maximal ideal of the local ring {\mathscr{O}}_s). Then (by (iv) of Proposition 2) X_s = X\times_S\{s\} \simeq \operatorname{Specan}({\mathscr{A}}_{\{s\}}) = \operatorname{Specan}({\mathscr{A}}(s)) where {\mathscr{A}}(s)={\mathscr{A}}_s/{\mathfrak{m}}_s{\mathscr{A}}_s={\mathscr{A}}_s\otimes_{{\mathscr{O}}_s}k={\mathscr{A}}_{\{s\}}; if on a neighbourhood U of s we have that {\mathscr{A}}|U \simeq \frac{{\mathscr{O}}_U[t_1,\ldots,t_n]}{(f_1,\ldots,f_m)} then {\mathscr{A}}(s) \simeq \frac{k[t_1,\ldots,t_n]}{(f_1(s),\ldots,f_m(s))}. X_s=\operatorname{Specan}({\mathscr{A}}(s)) is the closed subspace of \{s\}\times{\mathscr{E}}^n\simeq{\mathscr{E}}^n defined by the “equations” f_i(s)(z_1,\ldots,z_n) = 0 \qquad\text{for }i=1,2,\ldots,m.
More precisely, if the functor F_{\mathscr{A}} is represented in \mathsf{An}/S by the pair (X,\xi), then the functor F_{{\mathscr{A}}(s)}=F_{{\mathscr{A}}/\{s\}} is represented in \mathsf{An}/\{s\} by the pair (X_s,\xi_s), where X_s is the fibre of X at s and where \xi_s\colon{\mathscr{A}}_{X_s}\to{\mathscr{O}}_{X_s} is obtained from \xi\colon{\mathscr{A}}_X\to{\mathscr{O}}_X by reduction modulo {\mathfrak{m}}_s. In particular, the map from \operatorname{Hom}_{\{s\}}(\{s\},X_s) to \operatorname{Hom}_k({\mathscr{A}}(s),k) that sends a section f of X_s/\{s\} to the composite homomorphism f^*\circ\xi'_{s,f(s)} (where \xi'_{s,f(s)}\colon{\mathscr{A}}(s)\to{\mathscr{O}}_{X_{s,f(s)}} is induced by the germ of \xi_s at f(s)) is bijective. Since \operatorname{Hom}(\{s\},X_s) can be identified with the set of points of X_s by sending f\colon\{s\}\to X_s to the point f(s)=x (so that f^*=\epsilon_X\colon{\mathscr{O}}_{X_s,x}\to k is the augmentation homomorphism), we obtain a bijection between the set of points of X_s and the set of ideals of {\mathscr{A}}(s) that give k as their quotient, by sending x\in X_s to the inverse image under \xi'_{s,x}\colon{\mathscr{A}}(s)\to{\mathscr{O}}_{X_s,x} of the maximal ideal of {\mathscr{O}}_{X_s,x}. But {\mathscr{O}}_{X_s,x}={\mathscr{O}}_{X,x}/{\mathfrak{m}}_s{\mathscr{O}}_{X,x} (Exposé 10, §2), {\mathscr{A}}(s)={\mathscr{A}}_s/{\mathfrak{m}}_s{\mathscr{A}}_s, and \xi'_{s,x} is obtained by reduction modulo {\mathfrak{m}}_s of \xi'_X\colon{\mathscr{A}}_s\to{\mathscr{O}}_{X,x} (induced by the germ of \xi at x). We can thus state:
We obtain a bijection between the set of points of X_s and the set of maximal ideals {\mathfrak{n}} of {\mathscr{A}}_s such that {\mathfrak{n}}\supset{\mathfrak{m}}_s{\mathscr{A}}_s and {\mathscr{A}}_s/{\mathfrak{n}}\xleftarrow{\sim}k, by associating to each x\in X_s the inverse image under the homomorphism \xi'_x\colon{\mathscr{A}}_s\to{\mathscr{O}}_{X,x} of the maximal ideal of {\mathscr{O}}_{X,x}.
If {\mathfrak{n}} is the maximal ideal of {\mathscr{A}}_s that corresponds to the point x\in X_s, then we thus have a factorisation of \xi'_x as {\mathscr{A}}_s\to({\mathscr{A}}_s)_{\mathfrak{n}}\xrightarrow{\varphi_x}{\mathscr{O}}_{X,x}.
The homomorphism \widehat{\varphi}_x\colon\widehat{({\mathscr{A}}_s)_{\mathfrak{n}}}\to\widehat{{\mathscr{O}}}_{X,x} induced by \varphi_x by passing to completions of the local rings in question in an isomorphism.
Proof. We use the notation used in the proof of Proposition 1. We can write {\mathscr{A}}_s \simeq {\mathscr{O}}_s[t_1,\ldots,t_n]/{\mathscr{I}}_s where {\mathscr{I}}=(f_{1,s},\ldots,f_{n,s}), and then {\mathscr{O}}_{X,x} = {\mathscr{O}}_{S\times{\mathscr{E}}^n,x}/{\mathscr{I}}_x where {\mathscr{I}}=\eta_x({\mathscr{I}}_s{\mathscr{O}}_{S\times{\mathscr{E}}^n,x}[t_1,\ldots,t_n]); \eta_x is the germ of the homomorphism \eta\colon{\mathscr{O}}_{S\times{\mathscr{E}}^n}[t_1,\ldots,t_n]\to{\mathscr{O}}_{S\times{\mathscr{E}}^n} defined by \eta(t_i)=q^*(z_1) (for i=1,2,\ldots,n), and \xi_x is induced from \eta_x by passing to quotients, so that we have a commutative diagram with exact horizontal lines \begin{CD} 0 @>>> {\mathscr{I}}_s @>>> {\mathscr{O}}_s[t_1,\ldots,t_n] @>>> {\mathscr{A}}_s @>>> 0 \\@. @VVV @VV{\eta'_x}V @VV{\xi'_x}V \\0 @>>> {\mathscr{I}}_x @>>> {\mathscr{O}}_{S\times{\mathscr{E}}^n} @>>> {\mathscr{O}}_{X,x} @>>> 0 \end{CD} where \eta'_x is induced by \eta_x.
We denote by {\mathfrak{p}} the inverse image under \eta'_x of the maximal ideal of {\mathscr{O}}_{S\times{\mathscr{E}}^n}; {\mathfrak{p}} is the ideal generated by {\mathfrak{m}}_s and t_1-x_1,\ldots,t_n-x_n, where x_i\in k denotes the value at x of the section q^*(z_i) of {\mathscr{O}}_{S\times{\mathscr{E}}^n} (for i=1,2,\ldots,n). We can write a commutative diagram with exact horizontal lines \begin{CD} 0 @>>> ({\mathscr{I}}_s)_{\mathfrak{p}} @>>> {\mathscr{O}}_s[t_1,\ldots,t_n]_{\mathfrak{p}} @>>> ({\mathscr{A}}_s)_{\mathfrak{n}} @>>> 0 \\@. @VVV @VV{\psi_x}V @VV{\varphi_x}V \\0 @>>> {\mathscr{I}}_x @>>> {\mathscr{O}}_{S\times{\mathscr{E}}^n,x} @>>> {\mathscr{O}}_{X,x} @>>> 0 \end{CD} We can take the completion of this diagram to get \big({\mathscr{O}}_s[t_1,\ldots,t_n]\big)_{\mathfrak{p}}^\wedge \xrightarrow{\sim} \widehat{{\mathscr{O}}}_s[[v_1,\ldots,v_n]] with this isomorphism being defined by sending t_i to x_i+v_i (for i=1,2,\ldots,n); we also have that \widehat{{\mathscr{O}}}_{S\times{\mathscr{E}}^n,x} \simeq \widehat{{\mathscr{O}}}_s\widehat{\otimes}_k \widehat{{\mathscr{O}}}_{{\mathscr{E}}^n,(x_1,\ldots,x_n)} \simeq \widehat{{\mathscr{O}}}_s[[v_1,\ldots,v_n]] with this isomorphism being defined by sending q^*(z_i)_x to x_i+v_i; we thus see that \widehat{\psi}_x is an isomorphism. Since {\mathscr{I}}_x=\eta_x({\mathscr{I}}{\mathscr{O}}_{S\times{\mathscr{E}}^n,x}[t_1,\ldots,t_n]) is generated by ({\mathscr{I}})_{\mathfrak{p}}, we also have an isomorphism \widehat{({\mathscr{I}}_s)}_{\mathfrak{p}}\xrightarrow{\sim}\widehat{{\mathscr{I}}}_x. The proposition then follows.
The morphism \widehat{f}\colon X\to(|S|,{\mathscr{A}}) of ringed spaces (defined via \xi) is flat, i.e. the functor {\mathscr{M}}\mapsto\widetilde{{\mathscr{M}}}=\widehat{f}^*({\mathscr{M}}) from the category of {\mathscr{A}}-modules to the category of {\mathscr{O}}_X-modules is exact.
3 Analytic spectrum of a finite algebra
Let S be an analytic space over k (a non-discrete complete valued field). We say that an {\mathscr{O}}_S-algebra {\mathscr{A}} is finite if it is a coherent sheaf of {\mathscr{O}}_S-modules.
Every finite {\mathscr{O}}_S-algebra {\mathscr{A}} is of finite presentation (as an algebra).
Proof. Let s\in S; the fibre {\mathscr{A}}_s is an {\mathscr{O}}_s-module of finite type, generated by elements \sigma_{i,s} (for 1\leqslant i\leqslant n) that are integral over {\mathscr{O}}_s; for each i, let F_{i,s}(\sigma_{i,s})=0 be an equation of integral dependence: F_{i,s}\in{\mathscr{O}}_s[t_i] is unitary polynomial (for i=1,\ldots,n). In a small enough neighbourhood U of s, there exist sections \sigma_i\in\Gamma(U,{\mathscr{O}}_S) , with germs \sigma_{i,s} (respectively) at s (for 1\leqslant i\leqslant n), that generate {\mathscr{A}}|U, as well as unitary polynomials F_i\in\Gamma(U,{\mathscr{O}}_S[t_i]), with germs F_{i,s} (respectively) at s (for 1\leqslant i\leqslant n), and such that F_i(\sigma_i)=0 for i=1,\ldots,n. The sheaf {\mathscr{B}}={\mathscr{O}}_U[t_1,\ldots,t_n]/(F_1,\ldots,F_n) is coherent, and we have a surjective homomorphism {\mathscr{B}}\to{\mathscr{A}}|U, since the \sigma_i generate {\mathscr{A}}|U; the kernel {\mathscr{I}} of this homomorphism if of finite type; we can find a neighbourhood V\subset U of s such that {\mathscr{I}}|V is generated by the sections \bar{G_1},\ldots,\bar{G_m}. If G_1,\ldots,G_m\in\Gamma(V,{\mathscr{O}}_S[t_1,\ldots,t_n]) lift \bar{G_1},\ldots,\bar{G_m}, then we have {\mathscr{A}}|V \xleftarrow{\sim} {\mathscr{O}}_V[t_1,\ldots,t_n] / (F_1,\ldots,F_n,G_1,\ldots,G_m).
As a consequence: every finite {\mathscr{O}}_S-algebra {\mathscr{A}} defines an analytic space X=\operatorname{Specan}({\mathscr{A}}) over S.
Let {\mathscr{A}} be a finite {\mathscr{O}}_S-algebra. If X=\operatorname{Specan}({\mathscr{A}}) and s\in S, then the fibre X_s has a finite number of points, in bijective correspondence with the maximal ideals {\mathfrak{n}} of {\mathscr{A}}_s such that {\mathscr{A}}_s/{\mathfrak{n}}\simeq k. If {\mathfrak{n}} is the maximal ideal that corresponds to a point x\in X_s, then we have an isomorphism \varphi_x\colon({\mathscr{A}}_s)_{\mathfrak{n}}\xrightarrow{\sim}{\mathscr{O}}_{X,x}.
Proof. We can use the statements of Proposition 3 and Proposition 4, by remarking that {\mathscr{A}}_s is a semi-local ring (since {\mathscr{A}}(s)={\mathscr{A}}_s/{\mathfrak{m}}_s{\mathscr{A}}_s is of finite rank over k, and thus Artinian; furthermore, every maximal ideal of {\mathscr{A}}_s contains {\mathfrak{m}}_s{\mathscr{A}}_s, by the Cohen–Seidenberg theorem). Also, \widehat{\varphi}_x\colon\widehat{({\mathscr{A}}_s)_{\mathfrak{n}}}\xrightarrow{\sim}{\mathscr{O}}_{X,x} and, since ({\mathscr{A}}_s)_{\mathfrak{n}} is separated, \varphi_x is injective; since {\mathscr{O}}_{X,x} is quasi-finite over {\mathscr{O}}_s (since ({\mathscr{A}}_s)_{\mathfrak{n}} is), it follows from the preparation theorem that {\mathscr{O}}_{X,x} is finite over {\mathscr{O}}_s; we thus conclude by the Nakayama lemma (or the Krull theorem) that \varphi_x is also surjective.
Let (X,x) be an analytic germ, and let f_1,\ldots,f_N be generators of an ideal of definition of {\mathscr{O}}_{X,x}. Consider (X,x) as a germ over ({\mathscr{E}}^n,0) via the morphism (X,x)\to({\mathscr{E}}^n,0) defined by (f_1,\ldots,f_n). There exists an open neighbourhood U of 0 in {\mathscr{E}}^n along with a finite {\mathscr{O}}_U-algebra {\mathscr{A}} such that {\mathscr{A}}_0\simeq{\mathscr{O}}_{X,x}; the spectrum X'=\operatorname{Specan}({\mathscr{A}}) has a single point x' over 0, and (X,x) is ({\mathscr{E}}^n,0)-isomorphic to (X',x').
Proof. This follows from Proposition 6 by using Corollary 1.4 (ii) from Exposé 13, the preparation theorem, and the following lemma.
Let S be an analytic space and s\in S; consider a finite {\mathscr{O}}_{S,s}-algebra A; then there exists an open neighbourhood U of s in S along with a finite {\mathscr{O}}_U-algebra {\mathscr{A}} such that {\mathscr{A}}_s\simeq A.
Proof.
We first construct a neighbourhood V of s and a coherent {\mathscr{O}}_V-module {\mathscr{F}} such that {\mathscr{F}}_s\simeq A; for this, we use a finite presentation {\mathscr{O}}_{S,s}^p\xrightarrow{u}{\mathscr{O}}_{S,s}^q\to A\to 0 of A; if (u_{ij}) is the matrix of u, there exists a neighbourhood V of s and sections f_{ij} of {\mathscr{O}}_s on V such that (f_{ij})_s=u_{ij} for every pair (i,j); the matrix (f_{ij}) defines a homomorphism {\mathscr{O}}_V^p\to{\mathscr{O}}_V^q whose cokernel is the desired sheaf {\mathscr{F}}.
Since {\mathscr{F}}\otimes_{{\mathscr{O}}_V}{\mathscr{F}} is coherent, there exists a neighbourhood W\subset V of s and a homomorphism ({\mathscr{F}}\otimes_{{\mathscr{O}}_V}{\mathscr{F}})|W\to{\mathscr{F}}|W whose germ at s is identified with the multiplication A\otimes_{{\mathscr{O}}_S}A\to A of the algebra A; we can then find a neighbourhood U\subset W of s on which this homomorphism endows {\mathscr{A}}={\mathscr{F}}|U with the structure of an {\mathscr{O}}_U-algebra.
Every analytic algebra is a Henselian local ring.
Proof. For the definition and the equivalent characterisations of Henselian local rings, we refer to the appendix. We will use here the following characterisation:
For a local ring A to be Henselian, it is necessary and sufficient that, for every finite A-algebra B and every maximal ideal {\mathfrak{n}} of B, the localisation B_{\mathfrak{n}} is finite over A.
When A is an analytic algebra, Lemma 1 and Proposition 6 show that B_{\mathfrak{n}} is an analytic algebra finite over A for every maximal ideal {\mathfrak{n}} of B such that B/{\mathfrak{n}}=k. Consider an arbitrary maximal idea {\mathfrak{n}} of B, and set K=B/{\mathfrak{n}}; this is a finite extension of k. We introduce the algebras A'=A\otimes_kK and B'=B\otimes_kK; so A' is an analytic algebraic over the field K, and B' is finite over A'. To the ideal {\mathfrak{n}} of K there corresponds a maximal ideal {\mathfrak{n}}' of B', given by the kernel of the homomorphism from B' to K that extends B\to K; from B'/{\mathfrak{n}}'=K, we deduce that B'_{{\mathfrak{n}}'} is finite over A', and thus also over A (since A' is finite over A). It then follows that B_{\mathfrak{n}} is finite over A, since the canonical homomorphism B_{\mathfrak{n}}\to B'_{\mathfrak{n}} is clearly injective (and A is Noetherian). This finishes the proof.
Let {\mathscr{A}} be a finite {\mathscr{O}}_S-algebra. The structure morphism f\colon\operatorname{Specan}({\mathscr{A}})\to S of its analytic spectrum is a closed map.
Since the property is local on S, we can immediately reduce to the case where {\mathscr{A}} is generated by sections \sigma_1,\ldots,\sigma_n subject to relations F_i(\sigma_i)=0, where the F_i are unitary polynomials belonging to \Gamma({\mathscr{O}}_S)[t_i] (for i=1,\ldots,n), and G_j(\sigma_1,\ldots,\sigma_n)=0 (for j=1,\ldots,m), so that {\mathscr{A}}\simeq{\mathscr{B}}/{\mathscr{J}} with {\mathscr{B}}={\mathscr{O}}_S[t_1,\ldots,t_n]/(F_1,\ldots,F_n) (cf. the proof of Proposition 5). We then have a closed immersion \operatorname{Specan}({\mathscr{A}})\to\operatorname{Specan}({\mathscr{B}}), and it evidently suffices to prove the property for the structure morphism of \operatorname{Specan}({\mathscr{B}}); this follows from the following statement:
Let (x_p) be a sequence of points of \operatorname{Specan}({\mathscr{B}}) such that s_p=f(x_p) converges to a point s\in S. Then there exists a subsequence of (x_p) that converges.
By noting that \operatorname{Specan}({\mathscr{B}}) \simeq \operatorname{Specan}({\mathscr{O}}_S[t]/(F_1)) \times_S \ldots \times_S \operatorname{Specan}({\mathscr{O}}_S[t]/(F_n)) by Proposition 2, (ii), we see that it suffices to prove this statement in the case where n=1. For \operatorname{Specan}({\mathscr{O}}_S[t]/(F))\subset S\times{\mathscr{E}}, the result comes from the following:
Let