Lemma 38.2.3. Let X \to T \to S be morphisms of schemes with T \to S étale. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let x \in X be a point. Then
In particular \mathcal{F} is flat over S if and only if \mathcal{F} is flat over T.
Lemma 38.2.3. Let X \to T \to S be morphisms of schemes with T \to S étale. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Let x \in X be a point. Then
In particular \mathcal{F} is flat over S if and only if \mathcal{F} is flat over T.
Proof. As an étale morphism is a flat morphism (see Morphisms, Lemma 29.36.12) the implication “\Leftarrow ” follows from Algebra, Lemma 10.39.4. For the converse assume that \mathcal{F} is flat at x over S. Denote \tilde x \in X \times _ S T the point lying over x in X and over the image of x in T in T. Then (X \times _ S T \to X)^*\mathcal{F} is flat at \tilde x over T via \text{pr}_2 : X \times _ S T \to T, see Morphisms, Lemma 29.25.7. The diagonal \Delta _{T/S} : T \to T \times _ S T is an open immersion; combine Morphisms, Lemmas 29.35.13 and 29.36.5. So X is identified with open subscheme of X \times _ S T, the restriction of \text{pr}_2 to this open is the given morphism X \to T, the point \tilde x corresponds to the point x in this open, and (X \times _ S T \to X)^*\mathcal{F} restricted to this open is \mathcal{F}. Whence we see that \mathcal{F} is flat at x over T. \square
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