I am reading the proof of the long exact sequence involving $S$-class groups and $S$-units in Neukirch Algebraic Number Theory, Chapter I, Prop. 11.6, which states the following canonical sequence is exact: $$1\to \mathcal O\to \mathcal O(X) \to \prod_{\mathfrak p\in S} K^*/{\mathcal O_\mathfrak p^*} \to Cl(\mathcal O) \to Cl(\mathcal O(X)) \to 1,$$ where $\mathcal O$ is a ring of integers, $S$ is a set of nonzero prime ideals of $\mathcal O$, $X=Spec(\mathcal O)-S$, $\mathcal O(X)=(\mathcal O$ localized at the mulplicative set $\mathcal O-\bigcup_{\mathfrak{p}\in X}\mathfrak p$).
One crucial part is to show that if $X$ is a set of nonzero prime ideals of a Dedekind domain $\mathcal O$ that omits finitely many primes, and a nonzero prime $\mathfrak q$ is contained in $\bigcup_{\mathfrak{p}\in X}\mathfrak p$, then $\mathfrak q$ must be in $X$.
The Prime Avoidance Lemma only deals with the case where $X$ is finite (instead of cofinite). I saw some counterexamples (e.g. Can a prime in a Dedekind domain be contained in the union of the other prime ideals?) in a general Dedekind domain. I wonder if the long exact sequence still holds true in this case? If not, what if the ring is required to be a ring of integers?
