# Understanding Remmert-Stein extension theorem

I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.

A preliminary result is stated in various books (Stanislaw lojasiewicz's Intro to Complex Analytic Geometry page 239 lemma 3, Grauert-Remmert's Coherent Analytic Sheaves page 181, Fritzsche-Grauert's From Holomorphic functions to complex Manifolds page 150 6.9) as follows:

Proposition: Let $D$ be a neighbourhood of $0$ in $\mathbb{C}^n$, let $N$ be a subspace of $\mathbb{C}^n$ of dimension $<p$, where $0<p\leq n$. If $V$ is an analytic subset of $(D - N)$ of constant dimension $p$ then $\overline{V}$ is an analytic subset of $D$.

However, I have in my mind the example in $\mathbb{C}^2$ where $N=\{0\}$, $V=\bigcup Z(y-nx)$ in $\mathbb{C}^2-\{(0,0)\}$. The closure doesn't seem to be an analytic variety in any neighbourhood of $0$. What am I missing?

This result is also stated, although slightly differently (So that my example doesn't apply), in Wikipedia (https://en.wikipedia.org/wiki/Remmert%E2%80%93Stein_theorem) and in Gunning-Rossi's Analytic Functions of Several Complex Variables (page 169 where they require $V$ to be irreducible). However, I do not believe the proof provided in Gunning-Rossi as it proceeds their proof of the Proper Mapping theorem which is impossible to follow (for me) and perhaps incorrect (See the errata on Prof. Gunning's webpage).

All help is appreciated.

• I assume that you are taking the union of $\text{Zero}(y-ax)$ as $a$ ranges over an infinite set of integers. The set $V$ is not a complex analytic subvariety of $D\setminus N.$ The set $V$ accumulates on the line $\text{Zero}(x)$. – Jason Starr Oct 24 '17 at 22:49
• D'oh! Thank you kind sir. If you leave this as an answer, I will upvote and accept. – P. Brown Oct 24 '17 at 23:14

## 1 Answer

I am posting my comment as an answer. I assume that you are taking the union of $\text{Zero}(y-ax)$ as $a$ ranges over an infinite set of integers. The set $V$ is not a complex analytic subvariety of $D\setminus N.$ The set $V$ accumulates on the line $\text{Zero}(x).$