Proposition 38.39.4. Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. Then the category fibred in groupoids
whose fibre category over $U$ is the category of finite locally free $\mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}_ U$-modules over $U$ is a stack in groupoids. Moreover, if $U$ is quasi-compact and quasi-separated, then $\mathcal{S}_ U$ is $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(U)$.
Proof. The final assertion is the content of Lemma 38.39.1. To prove the proposition we will check conditions (1), (2), and (3) of Lemma 38.37.13.
Condition (1) holds because by definition we have glueing for the Zariski topology.
To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Since $Y, X, X \times _ Y X$ are quasi-compact and quasi-separated, we can use the description of fibre categories given in the statement of the proposition. Then it is clearly enough to show that
is an equivalence (as this will imply the same for the colimits). This follows immediately from fppf descent of finite locally free modules, see Descent, Proposition 35.5.2 and Lemma 35.7.6.
Condition (3) is the content of Lemmas 38.39.2 and 38.39.3. $\square$
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