Intuition on an object in algebraic geometry

Question

Let $X$ be a smooth projective variety over a field $k$ of characteristic zero, and let $D$ be a simple normal crossing divisor inside $X$ having irreducible components $D_i$. Further let $x \in X$ be a closed point belonging to (at least) one of these components, say $D_0$.

How should I think of the elements in

$\Gamma(Spec(\mathcal{O}_{X, x})-D_0, \mathbb{G}_m)$?

What is the order at $x$ of such an element?

Thanks for your help!

Answer 1

In $\mathrm{Spec}(\mathcal{O}_{X,x})$, $D_0$ is given by one equation $f=0$. So $\Gamma(Spec(\mathcal{O}_{X, x})-D_0, \mathbb{G}_m)$ is just $\mathcal{O}_{X, x}^*[\dfrac{1}{f} ]$: that is, any element is of the form $uf^n$, where $u$ is invertible near $x$ and $n\in \Bbb{Z}$. Its order at $x$ is $n$.