Relatively prime elements of a local ring stay relatively prime in nearby local rings

Question

Let $M$ be a complex manifold, $V\subset M$ be an irreducible analytic hyper surface and $p,q\in V$. I wonder why relatively prime elements of $\mathcal{O}_{V,p}$ stay relatively prime in $\mathcal{O}_{V,q}$, where $q$ is close to $p$?

Answer 1

Up to choosing an open chart we may reduce to $(\mathbb{C}^n,0)$. Let $f,g$ be two relative prime germs of analytic function at $0$. This means that there are germs $r,s$ such that $rf+sg$ does not vanish at the origin. Since we have only four germs, we can take representatives in a common neighborhood $U$ of the origin (just take a representative of each one and intersect their domains).

By continuity, there exists an open subset $0 \in V \subset U$ such that $rf+sg$ has no zeros. For every $q\in V$ the germs of $f$ and $g$ at $q$ will be relative prime.

In the same fashion, one can adapt the argument to a hypersurface requiring only that $rf+sg\not\in \mathcal{I}(V)_p$ (the germ of ideal of $V$).