The Stacks project

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

  1. 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

  2. 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.

Proof. Assume (1) and let $U, V$ be as in (2). It follows from Lemma 37.57.1 that $\mathcal{O}_ S(V) \to \mathcal{O}_ X(U)$ is pseudo-coherent. Hence it suffices to prove that the property of a ring map being "of finite tor dimension" satisfies conditions (1)(a), (b), (c) of Morphisms, Definition 29.14.1. These properties follow from More on Algebra, Lemmas 15.66.11, 15.66.14, and 15.66.16. Some details omitted. $\square$


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