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The Stacks project

Lemma 38.27.3. Assume that X \to S is a smooth morphism of affine schemes with geometrically irreducible fibres of dimension d and that \mathcal{F} is a quasi-coherent \mathcal{O}_ X-module of finite presentation. Then F_ d = \coprod _{p = 0, \ldots , c} H_ p for some c \geq 0 with F_ d as in (38.20.11.1) and H_ p as in (38.20.7.2).

Proof. As X is affine and \mathcal{F} is quasi-coherent of finite presentation we know that \mathcal{F} can be generated by c \geq 0 elements. Then \dim _{\kappa (x)}(\mathcal{F}_ x \otimes \kappa (x)) in any point x \in X never exceeds c. In particular H_ p = \emptyset for p > c. Moreover, note that there certainly is an inclusion \coprod H_ p \to F_ d. Having said this the content of the lemma is that, if a base change \mathcal{F}_ T is flat in dimensions \geq d over T and if t \in T, then \mathcal{F}_ T is free of some rank r in an open neighbourhood U \subset X_ T of the unique generic point \xi of X_ t. Namely, then H_ r contains the image of U which is an open neighbourhood of t. The existence of U follows from More on Morphisms, Lemma 37.16.7. \square


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