# stacks/stacks-project

Fix statement theorem from 9a7776c

Thanks to Bhargav Bhatt for pointing this out.
 @@ -11950,13 +11950,16 @@ \section{Descent vector bundles in positive characteristic} $$is fully faithful with essential image described as follows. Let \mathcal{E} be a finite locally free \mathcal{O}_X-module. Assume Assume for all y \in Y there exists integers n_y, r_y \geq 0 such that$$ \mathcal{E}|_{X_{y, red}} \cong \mathcal{O}_{X_{y, red}}^{\oplus r_y} F^{n_y, *}\mathcal{E}|_{X_{y, red}} \cong \mathcal{O}_{X_{y, red}}^{\oplus r_y} $$for all y \in Y. Then for some n \geq 0 the nth frobenius power pullback F^{n, *}\mathcal{E} is the pullback of a finite locally free \mathcal{O}_Y-module. Then for some n \geq 0 the nth frobenius power pullback F^{n, *}\mathcal{E} is the pullback of a finite locally free \mathcal{O}_Y-module. \end{theorem} \begin{proof} @@ -12006,17 +12009,18 @@ \section{Descent vector bundles in positive characteristic} and X_{y, red} = \lim Z_{i, red}. By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation} we can find an i such that \mathcal{E}|_{Z_{i, red}} \cong \mathcal{O}_{Z_{i, red}}^{\oplus r_y}. F^{n_y, *}\mathcal{E}|_{Z_{i, red}} \cong \mathcal{O}_{Z_{i, red}}^{\oplus r_y}. Fix i. We have Z_{i, red} = \lim Z_{i, j} where Z_{i, j} \to Z_i is a thickening of finite presentation (Limits, Lemma \ref{limits-lemma-closed-is-limit-closed-and-finite-presentation}). Using the same lemma as before we can find a j such that \mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}. F^{n_y, *}\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}. We conclude that for each y \in Y there exists a morphism Y_y \to Y of finite presentation whose image contains y and a thickening Z_y \to Y_y \times_Y X such that \mathcal{E}|_{Z_y} \cong \mathcal{O}_{Z_y}^{\oplus r_y}. F^{n_y, *}\mathcal{E}|_{Z_y} \cong \mathcal{O}_{Z_y}^{\oplus r_y}. Observe that the image of Y_y \to Y is constructible (Morphisms, Lemma \ref{morphisms-lemma-chevalley}). Since Y is quasi-compact in the constructible topology @@ -12028,9 +12032,9 @@ \section{Descent vector bundles in positive characteristic}$$ of finite presentation such that $Y = \bigcup \Im(Y_a \to Y)$ set theoretically and such that for each $a \in \{1, \ldots, N\}$ there is a thickening there exist integers $n_a, r_a \geq 0$ and there is a thickening $Z_a \subset Y_a \times_Y X$ of finite presentation such that $\mathcal{E}|_{Z_a} \cong \mathcal{O}_{Z_a}^{\oplus r_a}$. $F^{n_a, *}\mathcal{E}|_{Z_a} \cong \mathcal{O}_{Z_a}^{\oplus r_a}$. \medskip\noindent Formulated in this way, the condition descends to an absolute @@ -12067,7 +12071,8 @@ \section{Descent vector bundles in positive characteristic} \ref{limits-lemma-descend-closed-immersion-finite-presentation} and \ref{limits-lemma-descend-surjective}). Since $Z_a = \lim Z_{i, a}$ we find that after increasing $i$ we may assume $\mathcal{E}_i|_{Z_{i, a}} \cong \mathcal{O}_{Z_{i, a}}^{\oplus r_a}$, see $F^{n_a, *}\mathcal{E}_i|_{Z_{i, a}} \cong \mathcal{O}_{Z_{i, a}}^{\oplus r_a}$, see Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}. Finally, after increasing $i$ one more time we may assume $\coprod Y_{i, a} \to Y_i$ is surjective by @@ -12136,7 +12141,8 @@ \section{Descent vector bundles in positive characteristic} for $Y' \times_Y X \to Y'$ and the restriction of $\mathcal{E}$ to $Y' \times_Y X$. Suppose that we find some object $\xi'$ in $\colim_F \textit{Vect}(Y')$ pulling back to the restriction of $\mathcal{E}$ to $X'$. pulling back to the restriction of $\mathcal{E}$ to $X'$ (viewed as an object of the colimit category). By induction on $\dim(Y)$ we can find an object $\xi''$ in $\colim_F \textit{Vect}(E)$ pulling back to the restriction of $\mathcal{E}$ to $E \times_Y X$. Then the fully faithfullness