On the irreducible components of globally defined semianalytic sets
Let \({\mathcal A}\) be a class of geometric objects, whatever it may be. The naive way to define a nonempty set \(X\in{\mathcal A}\) to be \({\mathcal A}\)-reducible is to say that there exist \(Y\subsetneq X\) and \(Z\subsetneq X\) in \({\mathcal A}\) such that \(X=Y\cup Z\). Of course, \(X\) is \({\mathcal A}\)-irreducible if it is not \({\mathcal A}\)-reducible. The \({\mathcal A}\)-irreducible components of \(X\in{\mathcal A}\) are the members of a family of \({\mathcal A}\)-irreducible sets \(X_i\in{\mathcal A}\) such that \(X_i\neq X_j\) if \(i\neq j\), \(X=\bigcup_{i}X_i\) and, whenever \(Y\in{\mathcal A}\) is \({\mathcal A}\)-irreducible, it contains a component \(X_i\) and \(Y\subset X\), then \(Y=X_i\). In a honest irreducibility theory, the family of irreducible components of a given object is unique. In such a case \(X\) is \({\mathcal A}\)-irreducible if it is its unique \({\mathcal A}\)-irreducible component. On the other hand in most situations we can attach to every \(X\in{\mathcal A}\) a ring \({\mathcal F}(X)\) of either \(\mathbb C\)-valued or \(\mathbb R\)-valued functions such that \(X\) is irreducible if and only if \({\mathcal F}(X)\) is an integral domain. The first example is given by the class \({\mathcal A}\) of algebraic subsets of \(\mathbb C^n\). As the ring \(\mathbb C[{ x}]\) of polynomial functions on \(\mathbb C^n\) is noetherian, the ideal \({\mathcal J}(X)\) of those polynomial functions vanishing on \(X\) is a finite intersection of prime ideals \({\mathfrak p}_1,\dots,{\mathfrak p}_r\). Let \(X_i:=\bigcap_{f\in{\mathfrak p}_i}f^{-1}(0)\). Then \(\{X_1,\dots,X_r\}\) is the family of irreducible components of \(X\). This interplay between algebra and geometry was succesfully employed to develop satisfactory theories of irreducible components for complex analytic sets and Stein spaces by H. Cartan, O. Forster, R. Remmert and K. Stein, and global real analytic sets, introduced by H. Cartan and studied by F. Bruhat and H. Whitney, in the fifties of the past century. Later on, G. Efroymson, T. Mostowski and J. J. Risler created, once the noetherianity of the ring \({\mathcal N}(M)\) of Nash functions on a Nash manifold \(M\) was proved, a similar theory of irreducible components of Nash sets. On the other hand, note that every semialgebraic set \(X\) with at least two points \(p\) and \(q\) can be written as \(X=Y\cup Z\), where \(Y:=X\setminus\{p\}\) and \(Z:=\{p\}\), and both \(Y\) and \(Z\) are proper semialgebraic subsets of \(X\). This is why the author of the paper under review and the reviewer introduced in [Int. J. Math. 23, No. 4, Article ID 1250031, 40 p. (2012; Zbl 1271.14087)] the notion of Nash function on a semialgebraic set \(X\) that generalizes the notion of Nash function on a Nash manifold. They constitute a ring \({\mathcal N}(X)\) and \(X\) is said to be irreducible if \({\mathcal N}(X)\) is an integral domain. This provides a very satisfactory theory of irreducible components of semialgebraic sets that can be seen as the germ of the article under review. However, the situation is far more complicated in the semianalytic case and indeed the first remarkable fact is that the author has found the right subclass of semianalytic sets admitting a nice theory concerning irreducibility. Recall that a subset \(Z\) of a real analytic manifold \(M\) is a \(C\)-analytic set if there exist analytic functions \(f,f_1,\dots,f_m\in{\mathcal O}(M)\) with \(Z:=\{x\in M:f_1(x)=0,\dots, f_m(x)=0\}\). The author introduced in his joint article with \textit{F. Acquistapace} et al. [Math. Ann. 366, No. 1--2, 613--654 (2016; Zbl 1371.14063)] the notion of \(C\)-semianalytic set. A subset \(X\) of a real analytic manifold \(M\) is a \(C\)-semianalytic set if each point \(x\in M\) has an open neighborhood \(U^x\) in \(M\) where \(X\cap U^x\) is a finite union of sets of the form \(\{y\in U^x:f(y)=0,\, g_1(y)>0,\dots, g_r(y)>0\}\) and \(f,g_i\in{\mathcal O}(M)\) are analytic functions on \(M\). In the paper under review the author defines a subset \(X\) of an analytic manifold \(M\) to be an amenable \(C\)-semianalytic set if it is a finite union of \(C\)-semianalytic sets of the type \(Z\cap U\), where \(Z\subset M\) is a \(C\)-analytic set and \(U\subset M\) is an open \(C\)-semianalytic set. An amenable \(C\)-semianalytic set enjoys a crucial property: its dimension coincides with the dimension of its Zariski closure, that is, the smallest \(C\)-analytic subset of \(M\) containing it. In the paper under review the author proves that amenable \(C\)-semianalytic sets admit a solid notion of irreducibility and a satisfactory theory of irreducible components. It is worthwihle mentioning that, as in the semialgebraic case, the naive geometric notion of irreducibility does not work in the amenable \(C\)-semianalytic setting. To be precise, let \(X\subset M\) be an amenable \(C\)-semianalytic subset of an analytic manifold \(M\) with at least two points \(p\) and \(q\). Pick an open \(C\)-semianalytic neighborhood \(W\) of \(p\) in \(M\) such that \(q\notin W\). Then \(X=Y\cup Z\) where \[ Y:=X\cap W\quad\text{and}\quad Z:=X\setminus\{p\} \] are amenable \(C\)-semianalytic sets strictly smaller than \(X\). Thus, every amenable \(C\)-semianalytic set with at least two points would be reducible and this is not admissible. This leads the author to introduce an algebraic approach. Namely, he attaches to each amenable \(C\)-semianalytic set \(X\) the ring \({\mathcal O}(X)\) of real valued functions on \(X\) that admit an analytic extension to an open neighborhood of \(X\) in \(M\), and he defines \(X\) to be irreducible if \({\mathcal O}(X)\) is an integral domain. One of the main results of the paper is Theorem 5.9. It is proved that given an amenable \(C\)-semianalytic subset of an analytic manifold \(M\) there exists a unique locally finite family \(\{X_i:i\in\mathbb N\}\) of irreducible amenable \(C\)-semianalytic sets whose union is \(X\), with \(X_i\neq X_j\) if \(i\neq j\) and such that given an irreducible amenable \(C\) semianalytic set \(Y\) with \(X_i\subset Y\subset X\), then \(Y=X_i\). Of course, \(\{X_i:i\in\mathbb N\}\) is the family of irreducible components of the amenable \(C\)-semianalytic set \(X\). Indeed, there exists a bijection between the irreducible components of \(X\) and the minimal prime ideals of \({\mathcal O}(X)\). The introduced notions of irreducibility and irreducible components of amenable \(C\)-semianalytic set enjoy the expected property: they generalize the known notions concerning other classes of geometric sets both in the real and complex settings, for algebraic and analytic sets, Nash sets and semialgebraic sets. The relation between irreducibility and connectedness deserves some few words. It is very well known that irreducible algebraic subsets of \(\mathbb C^m\) are connected. This is not longer true in the real setting. For example, the real algebraic curve \(X:=\{(x,y)\in\mathbb R^2:x^2+y^2=x^3\}\) is an irreducible real algebraic set whose connected components are the origin \(\{(0,0)\}\) and \(X\setminus\{(0,0)\}\). Let \(X\subset M\) be an amenable \(C\)-semianalytic set and let \(Z\) be its Zariski closure. Let \(({\widetilde Z},\sigma)\) be a Stein complexification of \(Z\) together with the anti-involution \(\sigma:{\widetilde Z}\to {\widetilde Z}\) whose set of fixed points is \(Z\). Let \((Y,\pi)\) be the normalization of \({\widetilde Z}\) and let \({\widehat\sigma}:Y\to Y\) be the anti-holomorphic involution induced by \(\sigma\) in \(Y\), which satisfies the equality \(\pi\circ{\widehat\sigma}=\sigma\circ\pi\). A remarkable result in the paper under review is Theorem 1.2, where the author proves that the amenable \(C\)-semianalytic set \(X\) is irreducible if and only if there exists a connected component \(T\) of \(\pi^{-1}(X)\) such that \(\pi(T)=X\). Compare this statement with Corollary 3.5 of the article on irreducible semialgebraic sets quoted above. This article is well-written, in clear and precise style. The proofs, which are not trivial, contain details enough to be studied without a big effort by specialists in the field. This article constitutes, in some sense, the final step in the long process of finding the right notion (and develop the fundamentals) of irreducibility and a satisfactory theory of irreducible components in the real semianalytic setting.
\(C\)-analytic and \(C\)-semianalytic sets
scientific article; zbMATH DE number 6618703
scientific article