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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
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