Lemma 34.7.4 (021P)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.7: The fppf topology
Lemma 34.7.4 (cite)

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Lemma 34.7.4. Let $T$ be an affine scheme. Let $\{ T_ i \to T\} _{i \in I}$ be an fppf covering of $T$ . Then there exists an fppf covering $\{ U_ j \to T\} _{j = 1, \ldots , m}$ which is a refinement of $\{ T_ i \to T\} _{i \in I}$ such that each $U_ j$ is an affine scheme. Moreover, we may choose each $U_ j$ to be open affine in one of the $T_ i$ .

Proof. This follows directly from the definitions using that a morphism which is flat and locally of finite presentation is open, see Morphisms, Lemma 29.25.10. $\square$

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