Theorem 3 (Hartshorne)

Let $A$ be a noetherian ring, $F$ a closed subset of $\mathrm{Spec}(A)$ and $U$ the complementary open set. Suppose that $\mathrm{prof}_F(A) \geqslant 2$. Then, for every connected component $Y$ of $\mathrm{Spec}(A)$, the set $Y \cap \overline{U}$ is connected and dense in $Y$.

Suppose first that $\mathrm{Spec}(A)$ is connected. By remark 1, $U$ is dense in $\mathrm{Spec}(A)$, and it remains to prove that it is connected. Reasoning by contradiction, suppose that there are given two disjoint open sets $U_0$ and $U_1$ of $\mathrm{Spec}(A)$, non-empty and with union $U$. Since the set $\mathrm{Ass}(A)$ is contained in $U$ by remark 1, it is the disjoint union of $\mathrm{Ass}(A) \cap U_0$ and $\mathrm{Ass}(A) \cap U_1$. By IV, § 1, No. 1, Prop. 4, there exist ideals $J_0$ and $J_1$ of $A$ such that $\mathrm{Ass}(J_i) = \mathrm{Ass}(A) \cap U_i$, $\mathrm{Ass}(A/J_i) = \mathrm{Ass}(A) \cap U_{1-i} \quad (i = 0, 1)$. The complement of $U_i$ in $\mathrm{Spec}(A)$ contains $\mathrm{Ass}(A/J_i)$ and $\mathrm{Ass}(J_{1-i})$; since it is closed, it contains $\mathrm{Supp}(A/J_i)$ and $\mathrm{Supp}(J_{1-i})$. For $p \in U_i$, one has thus $(J_i)_p = A_p$ and $(J_{1-i})_p = 0$; this implies in particular that $J_0$ and $J_1$ are distinct from $A$. Let $B$ be the $A$-module $A/J_0 \times A/J_1$ and let $u : A \to B$ be the canonical homomorphism. By what precedes, the homomorphism $u_p$ is bijective for every $p \in U$; moreover, one has $\mathrm{Ass}(B) \subset U_0 \cup U_1 = U$, hence $\mathrm{prof}_F(B) \geqslant 1$ by remark 1. Lemma 4 then implies that $u$ is bijective, which contradicts the connectedness of $\mathrm{Spec}(A)$.

Let us treat the general case. Let $J$ be an ideal of $A$ such that $F = V(J)$ and let $Y$ be a connected component of $\mathrm{Spec}(A)$. By II, § 4, No. 3, Prop. 15, there exists an idempotent element $f$ of $A$ such that $Y$ is identified with the subset $\mathrm{Spec}(A_f)$ of $\mathrm{Spec}(A)$. Then $Y \cap F$ is identified with $V(J_f)$; one has $\mathrm{prof}_{A_f}(J_f, A_f) \geqslant \mathrm{prof}_A(J; A) \geqslant 2$ by Prop. 6, a) of No. 3. It follows from the first part of the proof that $Y \cap U = Y - (Y \cap F)$ is connected and dense in $Y$.