Global construction of the normalization of Stein spaces
Let X be a Stein space, \(\tilde X\) its normalization. Stein spaces X can be completely described by the algebra \({\mathcal O}(X)\) of global holomorphic functions. The question is can the normalization \(\tilde X\) of X be constructed just from the holomorphic functions on X. Or equivalently whether the algebra \(\tilde {\mathcal O}(X)\) of all weakly holomorphic functions on X can be derived from the algebra \({\mathcal O}(X)\) of holomorphic functions on X. The main results of the paper are as follows: if X is irreducible \(\tilde {\mathcal O}(X)\) is the topological closure of the integral closure \(\tilde {\mathcal O}(X)\) of \({\mathcal O}(X)\). An example is given when \(\tilde {\mathcal O}(X)\) is not topologically closed and X is locally irreducible. \(\tilde {\mathcal O}(X)\) can also be obtained by taking the intersection of the localizations \(S_ x^{-1}\tilde {\mathcal O}(X)\) of \(\tilde {\mathcal O}(X)\) with respect to \(S_ x:=\{g\in {\mathcal O}(X):\) g(x)\(\neq 0\}\) for every \(x\in X\).
scientific article; zbMATH DE number 4107182