The Weierstrass Preparation Theorem and Its Consequences

Let Z be a ℂ $${\mathbb C\mskip 1mu}$$ -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 ( φ , φ ∗ ) : ( U , O Z ∕ U ) → ( V , O V ) $$(\varphi ,\varphi ^*): (U,\mathbb {O}_Z/U) \to (V,\mathbb {O}_V)$$ , where U is an open neighborhood of x in Z, V is an open neighborhood of 0 in some ℂ n $${\mathbb C\mskip 1mu}^n$$ , and O V $$\mathbb {O}_V$$ is the sheaf of holomorphic functions on V such that φ(x) = 0. If h ∈ O V ( V ′ ) $$h \in \mathbb {O}_V (V')$$ 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 ℂ n $${\mathbb C\mskip 1mu}^n$$ , we call the sections ( z 1 , … , z n ) ∈ O Z ( U ) n $$(z_1,\dots ,z_n) \in \mathbb {O}_Z(U)^n$$ 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).