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 complete valued non-discrete 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{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}(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, Section 1).
- {\mathscr{A}}\mapsto\operatorname{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\mathscr{A}}\otimes_{{\mathscr{O}}_S}{\mathscr{B}})\simeq\operatorname{Spec}_\mathrm{an}({\mathscr{A}})\times_S\operatorname{Spec}_\mathrm{an}({\mathscr{B}}).
- Let h\colon{\mathscr{A}}\to{\mathscr{B}} be a surjective homomorphism of algebras of finite presentation. Then the corresponding morphism \operatorname{Spec}_\mathrm{an}(h)\colon\operatorname{Spec}_\mathrm{an}({\mathscr{B}})\to\operatorname{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\mathscr{A}})\times_S T\simeq\operatorname{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\mathscr{A}}). If s\in S, recall (Exposé 10, Section 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{Spec}_\mathrm{an}({\mathscr{A}}_{\{s\}}) = \operatorname{Spec}_\mathrm{an}({\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{Spec}_\mathrm{an}({\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,