Let $X$ be a scheme and $D$ be a closed principle subscheme, i.e. the ideal sheaf $\mathcal I_D$ that cuts out $D$ is locally a principal ideal.
The open complement $U=X\setminus D$ induces an open immersion $j: U \to X$ locally given by the localization of a generator of $\mathcal I_D$. In particular $j$ is affine.
How can we prove this local interpretation? Thanks for your help.
Edit Locally we assume $X=\operatorname{Spec}(A)$ affine and we cover it $X= \cup_i D(f_i)$ by distinguished open subsets for $f_i \in A, \forall i$. Then
$j^{-1}(D(f_i)) = D(f_i) = Spec (A/f_i)$. So we can see that $j$ is indeed affine but the description of $j$ is not given in term of localization by a generator of and ideal $I_D$, locally corresponding to the ideal sheaf $\mathcal I_D$?
Edit2 If we consider $D=\operatorname{Spec}(A/I)$ locally given by the ideal $I=\langle f \rangle\subseteq A$, then $U = \{\mathfrak p\in X=\operatorname{Spec A}| f\not\in\mathfrak p \} = D(f)$. Hence $j$ corresponds to the localization by $f$: $A \to A_f$.