Lemma 37.58.1. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. The following are equivalent

there exist an affine open covering $S = \bigcup V_ j$ and for each $j$ an affine open covering $f^{-1}(V_ j) = \bigcup U_{ji}$ such that $\mathcal{O}_ S(V_ j) \to \mathcal{O}_ X(U_{ij})$ is a perfect ring map, and

for every pair of affine opens $U \subset X$, $V \subset S$ such that $f(U) \subset V$ the ring map $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is perfect.

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