The Stacks project

Proof. Assume (1). Since excellent rings are quasi-excellent, it is clear that $X$ is quasi-excellent. Since excellent rings are universally catenary, it also follows that $X$ is universally catenary by Lemma 29.17.2. Assume (2). Let $x \in X$. Since $X$ is quasi-excellent there exists an affine open neighbourhood $x \in U \subset X$ such that $\mathcal{O}_ X(U)$ is quasi-excellent. Since $X$ is universally category, the ring $\mathcal{O}_ X(U)$ is also universally catenary by Lemma 29.17.2, whence excellent. $\square$