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Fix statement theorem from 9a7776c

Thanks to Bhargav Bhatt for pointing this out.
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aisejohan committed Sep 11, 2018
1 parent 9e9168e commit 3adba3238a083e0b8b294aab356b3af355990538
Showing with 18 additions and 12 deletions.
  1. +18 −12 flat.tex
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@@ -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}
$$
for all $y \in Y$. Then for some
$n \geq 0$ the $n$th frobenius power pullback $F^{n, *}\mathcal{E}$
is the pullback of a finite locally free $\mathcal{O}_Y$-module.
Then for some $n \geq 0$ the $n$th frobenius power pullback
$F^{n, *}\mathcal{E}$ is the pullback of a finite locally free
$\mathcal{O}_Y$-module.
\end{theorem}
\begin{proof}
@@ -12006,17 +12009,18 @@ \section{Descent vector bundles in positive characteristic}
and $X_{y, red} = \lim Z_{i, red}$.
By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}
we can find an $i$ such that
$\mathcal{E}|_{Z_{i, red}} \cong \mathcal{O}_{Z_{i, red}}^{\oplus r_y}$.
$F^{n_y, *}\mathcal{E}|_{Z_{i, red}} \cong
\mathcal{O}_{Z_{i, red}}^{\oplus r_y}$.
Fix $i$.
We have $Z_{i, red} = \lim Z_{i, j}$ where $Z_{i, j} \to Z_i$
is a thickening of finite presentation (Limits, Lemma
\ref{limits-lemma-closed-is-limit-closed-and-finite-presentation}).
Using the same lemma as before we can find a $j$ such that
$\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}$.
$F^{n_y, *}\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}$.
We conclude that for each $y \in Y$ there exists a morphism
$Y_y \to Y$ of finite presentation whose image contains $y$
and a thickening $Z_y \to Y_y \times_Y X$ such that
$\mathcal{E}|_{Z_y} \cong \mathcal{O}_{Z_y}^{\oplus r_y}$.
$F^{n_y, *}\mathcal{E}|_{Z_y} \cong \mathcal{O}_{Z_y}^{\oplus r_y}$.
Observe that the image of $Y_y \to Y$ is constructible
(Morphisms, Lemma \ref{morphisms-lemma-chevalley}).
Since $Y$ is quasi-compact in the constructible topology
@@ -12028,9 +12032,9 @@ \section{Descent vector bundles in positive characteristic}
$$
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

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