The Weierstrass Preparation Theorem and Its Consequences

Abstract

Let Z be a \({\mathbb C\mskip 1mu}\)-space and let x ∈ Z be a smooth point (see Definition 1.1.7 in Chap. 1). To take a local coordinate system on Z centered at x is, by definition, to consider a specific isomorphism \((\varphi ,\varphi ^*): (U,\mathbb {O}_Z/U) \to (V,\mathbb {O}_V)\), where U is an open neighborhood of x in Z, V is an open neighborhood of 0 in some \({\mathbb C\mskip 1mu}^n\), and \(\mathbb {O}_V\) is the sheaf of holomorphic functions on V such that φ(x) = 0. If \(h \in \mathbb {O}_V (V')\) is any holomorphic function on the open subset V ′⊂ V , we denote again by h the pull-back function φ∗(h). In particular, if (z1, …, zn) = z is the standard coordinate system on \({\mathbb C\mskip 1mu}^n\), we call the sections \((z_1,\dots ,z_n) \in \mathbb {O}_Z(U)^n\) the (corresponding) local coordinates on Z centered at x. We will also say, more succinctly, that we are taking a local coordinate system (z1, …, zn) = z on Z centered at x (or around x).