## On the fibers of analytic mappings.(English)Zbl 0792.13005

Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 45-101 (1993).
Let $$f:X \to S$$ be a morphism of complex or real spaces, $$P$$ a property of morphisms of local rings and $$P(f)$$ the set of $$x \in X$$ such that the induced map $${\mathcal O}_{S,f(x)} \to {\mathcal O}_{X,x}$$ has property $$P$$. Here a criterion is given for $$P(f)$$ to be constructible (resp. Zariski open) in $$X$$. More precisely, let $${\mathcal R}_{\mathbb{Q}}$$ be the category of all locally $$\mathbb{Q}$$-ringed spaces with Noetherian stalks, $$p:{\mathcal B} \to {\mathcal R}_ \mathbb{Q}$$ a fibered category over $${\mathcal R}_ \mathbb{Q}$$ and $$P$$ a property of the triple $$(C/A,G)$$, where $$A \to C$$ is a local morphism of local Noetherian $$\mathbb{Q}$$-algebras and $$G \in {\mathcal B} (C)$$. Let $$f:X \to S$$ be a morphism in $${\mathcal R}_ \mathbb{Q}$$, $$G \in {\mathcal B} (X)$$ and $$P(X/S;G)$$ the set of points $$x \in X$$ for which $$P({\mathcal O}_{X,x}/{\mathcal O}_{S,f(x)};G_ x)$$ holds. A property $$P$$ is constructible (resp. open) if for every morphism $$f:X \to S$$ of finite dimensional excellent Noetherian $$\mathbb{Q}$$-schemes and every subscheme $$Y \subset X$$ of finite type over $$S$$ such that there exists an $$f$$-regular module $$E$$ for $$Y$$ [i.e. for every $$y \in Y$$ and every generalization $$x \in X(s)$$; $$s:=f(y)$$ of $$y$$ the ring $${\mathcal O}_{X(s),x}$$ is regular iff $$E(s)_ x$$ is a free $${\mathcal O}_{X(s),x}$$-module] the set $$P(X/S;G) \cap Y$$ is constructible (resp. open) in $$Y$$ for every $$G \in {\mathcal B} (X)$$. A property $$P$$ is analytic if it is invariant under completion and satisfies the following assumptions:
(i) (Invariance under regular base) Let $$k$$ be a field with $$\text{char} (k)=0$$, $$A \to A'$$, $$A \to C$$ complete local analytic $$k$$-algebras such that $$A'$$ is regular over $$A$$ and $$C'$$ the localization of $$C \widehat {\bigotimes}_ A A'$$ with respect to some maximal ideal. Then for each $$G \in {\mathcal B} (C)$$, $$P(C/A,G)$$ iff $$P(C'/A',G \bigotimes_ CC')$$.
(ii) Let $$X$$ be a formal complex or formal real space and $$x \in X$$. Then the natural functor $$\varinjlim_ K {\mathcal B} (X \langle K \rangle) \to \varinjlim_ K {\mathcal B} (X| K)$$ is essentially surjective, where $$K$$ runs through the Stein compact sets $$K \subset X$$ with $$x \in K^ 0$$, $$X| K:=(K,{\mathcal O}_ X| K)$$, $$X \langle K \rangle:=\text{Spec} (\Gamma (K,{\mathcal O}_ X))$$ and $$X| K \to X \langle K \rangle$$ is the canonical morphism. The main result of this paper says that if $$P$$ is constructible (resp. open) and analytic and $$G \in {\mathcal B} (X)$$ then $$P(X/S;G)$$ is constructible (resp. open) in $$X$$.
A property $$P$$ is absolutely constructible if: (a) for every subfield $$k$$ of a local Noetherian $$\mathbb{Q}$$-algebra $$C$$ and $$G \in {\mathcal B} (C)$$ it holds $$P(C/k;G)$$ iff $$P(C/\mathbb{Q};G)$$; (b) $$P(X/ \mathbb{Q};G)$$ is constructible in $$X$$ for every excellent $$\mathbb{Q}$$-scheme $$X$$ and $$G \in {\mathcal B} (X)$$.
Another result of this paper says that if $$P$$ is absolutely constructible then the generic principle holds, i.e. if the structure morphism $$X \to \text{Spec} \mathbb{C}$$ (resp. $$X \to \text{Spec} \mathbb{R})$$ of a complex (resp. real) space $$X$$ has the property $$P$$ and if the regular locus $$\text{Reg} (S)$$ is dense in $$S$$, then for a dense set of parameters $$s \in S$$, property $$P$$ holds for $$f$$ in the point of the fiber $$f^{-1} (s)$$.
As applications the authors obtain the classical theorem of Sard for $$P=$$ “smooth”, the Frisch theorem on generic flatness for $$P=$$ “flat” and the fact that a family of complex spaces admits generically a simultaneous resolution or normalization. The paper contains a lot of examples where the properties are from commutative algebra as $$(R_ n)$$, $$(S_ n)$$, normal, Gorenstein and others.
For the entire collection see [Zbl 0772.00007].

### MSC:

 13B10 Morphisms of commutative rings 13H99 Local rings and semilocal rings 13J07 Analytical algebras and rings 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14D99 Families, fibrations in algebraic geometry 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 32B05 Analytic algebras and generalizations, preparation theorems