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Add in all the missing details from 9a7776c

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aisejohan committed Sep 10, 2018
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217 flat.tex
@@ -11791,7 +11791,7 @@ \section{Descent vector bundles in positive characteristic}
corresponding map $L \to A^{\oplus r}$ has the desired property.
Observe that $K \otimes_A Q(A) \cong L \otimes_A Q(A)$
by flat base change and the assumption that $\mathcal{E}$
is free on the generic fibre. Observe that
is free on the generic fibre. Let $\pi \in A$ be a uniformizer. Observe that
$$
K \otimes_A^\mathbf{L} A/\pi^m A =
R\Gamma(X, \mathcal{E} \xrightarrow{\pi^m} \mathcal{E})
@@ -11902,12 +11902,12 @@ \section{Descent vector bundles in positive characteristic}
the pullback of this to $X \times_S \Spec(\mathcal{O}_{S, s})$
is an isomorphism for every codimension $1$ point $s \in S$.
By Divisors, Lemma \ref{divisors-lemma-check-isomorphism-via-depth-and-ass}
it suffices to prove that $\text{depth} (f^*f_*\mathcal{E})_x \geq 2$
it suffices to prove that $\text{depth}((f^*f_*\mathcal{E})_x) \geq 2$
for any point $x \in X$ mapping to a point $s \in S$ of codimension $\geq 2$.
Since $f$ is flat and
$(f^*f_*\mathcal{E})_x = (f_*\mathcal{E})_s \otimes_{\mathcal{O}_{S, s}}
\mathcal{O}_{X, x}$, it suffices to prove that
$\text{depth} (f_*\mathcal{E})_s \geq 2$, see
$\text{depth}((f_*\mathcal{E})_s) \geq 2$, see
Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module}.
Since $S$ is a normal Noetherian scheme
and $\dim(\mathcal{O}_{S, s}) \geq 2$
@@ -11978,7 +11978,9 @@ \section{Descent vector bundles in positive characteristic}
and denote $g_1$ and $g_2$ the two pullbacks of $g$ to $X \times_Y X$.
Since $X_{y, red}$ is geometrically connected, we
see that $H^0(X_{y, red}, \mathcal{O}_{X_{y, red}})$ is
a purely inseparable extension of $\kappa(y)$.
a purely inseparable extension of $\kappa(y)$, see
Varieties, Lemma
\ref{varieties-lemma-proper-geometrically-reduced-global-sections}.
Thus $g^q|_{X_{y, red}}$ comes from an element of
$\kappa(y)$ for some $p$-power $q$ (which may depend on $y$).
It follows that $g_1^q$ and $g_2^q$ map to the same
@@ -11989,80 +11991,203 @@ \section{Descent vector bundles in positive characteristic}
to be a $p$-power as desired.
\medskip\noindent
Let $\mathcal{E}$ be as in the statement of the proposition.
Let us first massage the condition a bit. Namely, we can write the morphism
$y \to Y$ as a filtered colimit of immersions $Y_i \to Y$ of
finite presentation with $Y_i$ affine. For each $i$ set
$Z_i = Y_i \times_Y X$. Then $X_{y, red} = \lim Z_{i, red}$.
Moreover, we have $Z_{i, red} = \lim Z_{i, j}$ where
$Z_{i, j} \to Z_i$ is a thickening of finite presentation.
Description of essential image. Let $\mathcal{E}$ be as in the statement
of the proposition. We first reduce to the Noetherian case.
\medskip\noindent
Let $y \in Y$ be a point and view it as a morphims
$y \to Y$ from the spectrum of the residue field into $Y$.
We can write $y \to Y$ as a filtered limit of morphisms $Y_i \to Y$ of
finite presentation with $Y_i$ affine. (It is best to prove this
yourself, but it also follows formally from
Limits, Lemma \ref{limits-lemma-relative-approximation} and
\ref{limits-lemma-limit-affine}.)
For each $i$ set $Z_i = Y_i \times_Y X$. Then $X_y = \lim Z_i$
and $X_{y, red} = \lim Z_{i, red}$.
By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}
we can find a pair $(i, j)$ such that
we can find an $i$ such that
$\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}$.
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}$.
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
and since the assumption holds for all $y \in Y$,
we conclude we can find a finite number of immersions
(Topology, Lemma \ref{topology-lemma-constructible-hausdorff-quasi-compact} and
Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral})
we conclude that there find a finite number of morphisms
$$
Y_1 \to Y, Y_2 \to Y, \ldots, Y_n \to Y
Y_1 \to Y,\ Y_2 \to Y,\ \ldots,\ Y_N \to Y
$$
of finite presentation such that $Y$ is contained in $\bigcup Y_i$
set theoretically and such that for each $i$ there is a thickening
$Z_i \subset Y_i \times_Y X$ of finite presentation such that
$\mathcal{E}|_{Z_i} \cong \mathcal{O}_{Z_i}^{\oplus r_i}$.
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
$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}$.
\medskip\noindent
Formulated in this way, the condition descends to an absolute
Noetherian approximation. Details omitted. This reduces us to
the case discussed in the next section.
\medskip\noindent
Assume $Y$ is of finite type over $\mathbf{F}_p$. By the fully faithfulness
already proven and because of
Noetherian approximation. We stronly urge the reader to skip
this paragraph. First write $Y = \lim_{i \in I} Y_i$ as a cofiltered limit
of schemes of finite type over $\mathbf{F}_p$ with affine transition
morphisms (Limits, Lemma \ref{limits-lemma-relative-approximation}).
Next, we can assume we have proper morphisms $f_i : X_i \to Y_i$
whose base change to $Y$ recovers $f : X \to Y$, see
Limits, Lemma \ref{limits-lemma-descend-finite-presentation}.
After increasing $i$ we may assume there exists a finite locally
free $\mathcal{O}_{X_i}$-module $\mathcal{E}_i$ whose pullback
to $X$ is isomorphic to $\mathcal{E}$, see
Limits, Lemma \ref{limits-lemma-descend-invertible-modules}.
Pick $0 \in I$ and denote $E \subset Y_0$ the constructible subset
where the geometric fibres of $f_0$ are connected, see
More on Morphisms, Lemma
\ref{more-morphisms-lemma-nr-geom-connected-components-constructible}.
Then $Y \to Y_0$ maps into $E$, see
More on Morphisms, Lemma
\ref{more-morphisms-lemma-base-change-fibres-geometrically-connected}.
Thus $Y_i \to Y_0$ maps into $E$ for $i \gg 0$, see
Limits, Lemma \ref{limits-lemma-limit-contained-in-constructible}.
Hence we see that the fibres of $f_i$ are geometrically connected
for $i \gg 0$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation}
for large enough $i$ we can find morphisms
$Y_{i, a} \to Y_i$ of finite type whose base change to $Y$
recovers $Y_a \to Y$, $a \in \{1, \ldots, N\}$.
After possibly increasing $i$ we can find thickenings
$Z_{i, a} \subset Y_{i, a} \times_{Y_i} X_i$ whose base change
to $Y_a \times_Y X$ recovers $Z_a$ (same reference as before
combined with
Limits, Lemmas
\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
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
Limits, Lemma \ref{limits-lemma-descend-surjective}.
At this point all the assumptions hold for $X_i \to Y_i$
and $\mathcal{E}_i$ and we see that it suffices to prove result
for $X_i \to Y_i$ and $\mathcal{E}_i$.
\medskip\noindent
Assume $Y$ is of finite type over $\mathbf{F}_p$.
To prove the result we will use induction on $\dim(Y)$.
We are trying to find an object of
$\colim_F \textit{Vect}(Y)$ which pulls back to the
object of $\colim_F \textit{Vect}(X)$ determined by $\mathcal{E}$.
By the fully faithfulness already proven and because of
Proposition \ref{proposition-h-descent-vector-bundles-p}
it suffices to construct a descent of $\mathcal{E}$
after replacing $Y$ by the members of a h covering
and $X$ by the corresponding base change.
We may also replace $X$ by its reduction (same reason).
Consider the Stein factorization $X \to Y' \to Y$.
and $X$ by the corresponding base change. This means
that we may replace $Y$ by a scheme proper and surjective
over $Y$ provided this does not increase the dimension of $Y$.
If $T \subset T'$ is a thickening of schemes of finite type
over $\mathbf{F}_p$ then
$\colim_F \textit{Vect}(T) = \colim_F \textit{Vect}(T')$
as $\{T \to T'\}$ is a h covering such that $T \times_{T'} T = T$.
If $T' \to T$ is a universal homeomorphism of schemes
of finite type over $\mathbf{F}_p$, then
$\colim_F \textit{Vect}(T) = \colim_F \textit{Vect}(T')$
as $\{T \to T'\}$ is a h covering such that the diagonal
$T \subset T \times_{T'} T$ is a thickening.
\medskip\noindent
Using the general remarks made above, we may and do replace
$X$ by its reduction and we may assume $X$ is reduced.
Consider the Stein factorization $X \to Y' \to Y$, see
More on Morphisms, Theorem
\ref{more-morphisms-theorem-stein-factorization-Noetherian}.
Then $Y' \to Y$ is a universal homeomorphism
and hence we may replace $Y$ by $Y'$ (same reason again).
Thus we may assume $f_*\mathcal{O}_X = \mathcal{O}_Y$.
In particular $Y$ is reduced. In this case the morphism
$X \to Y$ is flat over a dense open subscheme $V \subset Y$, see
of schemes of finite type over $\mathbf{F}_p$.
By the above we may replace $Y$ by $Y'$.
Thus we may assume $f_*\mathcal{O}_X = \mathcal{O}_Y$
and that $Y$ is reduced. This reduces us to the case discussed
in the next paragraph.
\medskip\noindent
Assume $Y$ is reduced and $f_*\mathcal{O}_X = \mathcal{O}_Y$
over a dense open subscheme of $Y$.
Then $X \to Y$ is flat over a dense open
subscheme $V \subset Y$, see
Morphisms, Proposition \ref{morphisms-proposition-generic-flatness-reduced}.
By Lemma \ref{lemma-flat-after-blowing-up}
there is a $V$-admissible blowing up $Y' \to Y$ such that
the strict transform $X'$ of $X$ is flat over $Y'$.
By More on Morphisms, Lemma
Observe that $\dim(Y') = \dim(Y)$ as $Y$ and $Y'$ have
a common dense open subscheme. By More on Morphisms, Lemma
\ref{more-morphisms-lemma-proper-flat-nr-geom-conn-comps-lower-semicontinuous}
and the fact that $V \subset Y'$ is dense
all fibres of $f' : X' \to Y'$ are geometrically connected.
We still have $(f'_*\mathcal{O}_{X'})|_V = \mathcal{O}_V$.
Finally, we consider the morphism
Write
$$
Y' \times_Y X = X' \cup E \times_Y X
$$
where $E \subset Y'$ is the exceptional divisor of the blowing up.
By the general remarks above, it suffices to prove existence
for $Y' \times_Y X \to Y'$ and the restriction of $\mathcal{E}$
to $Y' \times_Y X$.
Suppose that we find some object $\xi'$ in $\colim_F \textit{Vect}(Y')$
pulling back to the restriction of $\mathcal{E}$ to $X'$.
By induction on $\dim(Y)$ we can find an object $\xi''$ in
$\colim_F \textit{Vect}(E)$ pulling back to the restriction of
$\mathcal{E}$ to $E \times_Y X$. Then the fully faithfullness
determines a unique isomorphism $\xi'|_E \to \xi''$
compatible with the given identifications with the restriction
of $\mathcal{E}$ to $E \times_{Y'} X'$. Since
$$
g : T = (Y')^\nu \times_{Y'} X' \longrightarrow (Y')^\nu = S
\{E \times_Y X \to Y' \times_Y X, X' \to Y' \times_Y X\}
$$
As before it suffices to show the statement for this one.
Observe that $g_*\mathcal{O}_T = \mathcal{O}_S$ (because it
is true over $V$ etc) and that $S$ is normal.
This reduces us to the case handled in the next paragraph.
is a h covering given by a pair of closed immersions with
$$
(E \times_Y X) \times_{(Y' \times_Y X)} X' = E \times_{Y'} X'
$$
we conclude that $\xi'$ pulls back to the restriction of
$\mathcal{E}$ to $Y' \times_Y X$. Thus it suffices to find
$\xi'$ and we reduce to the case discussed in the next paragraph.
\medskip\noindent
Assume $Y$ is reduced, $f$ is flat, and $f_*\mathcal{O}_X = \mathcal{O}_Y$
over a dense open subscheme of $Y$. In this case we consider the
normalization $Y^\nu \to Y$ (Morphisms, Section
\ref{morphisms-section-normalization}). This is a finite surjective
morphism
(Morphisms, Lemma \ref{morphisms-lemma-nagata-normalization} and
\ref{morphisms-lemma-ubiquity-nagata}) which is an isomorphism
over a dense open. Hence by our general remarks we may
replace $Y$ by $Y^\nu$ and $X$ by $Y^\nu \times_Y X$.
After this replacement we see that $\mathcal{O}_Y = f_*\mathcal{O}_X$
(because the Stein factorization has to be an isomorphism
in this case; small detail omitted).
\medskip\noindent
Assume $Y$ is a normal Noetherian scheme, that $f$ is flat, and
that $f_*\mathcal{O}_X = \mathcal{O}_Y$.
By the discussion in the third parapgraph,
we may assume there is a dense open subscheme
$V \subset Y$ such that $\mathcal{E}|_{f^{-1}(V)}$ is free.
that $f_*\mathcal{O}_X = \mathcal{O}_Y$. After replacing $\mathcal{E}$
by a suitable Frobenius power pullback, we may assume $\mathcal{E}$
is trivial on the scheme theoretic fibres of $f$ at the generic points
of the irreducible components of $Y$ (because
$\colim_F \textit{Vect}(-)$ is an equivalence on universal
homeomorphisms, see above). Similarly to the arguments above
(in the reduction to the Noetherian case) we conclude there is a dense
open subscheme $V \subset Y$ such that $\mathcal{E}|_{f^{-1}(V)}$ is free.
Let $Z \subset Y$ be a closed subscheme such that
$Y = V \amalg Z$ set theoretically. Let $\eta \in Y$ be the
generic point. Let $z_1, \ldots, z_t \in Z$
$Y = V \amalg Z$ set theoretically. Let $z_1, \ldots, z_t \in Z$
be the generic points of the irreducible components of $Z$
of codimension $1$. Then $A_i = \mathcal{O}_{Y, z_i}$ is
a discrete valuation ring. Let $n_i$ be the integer found in
Lemma \ref{lemma-trivial-fibres-dvr} for the scheme $X_{A_i}$ over $A_i$.
After replacing $\mathcal{E}$ by a suitable Frobenius
power pullback, we may assume $\mathcal{E}$ is free over
$X_{A_i/ \mathfrak m_i^{n_i}}$ (see arguments above).
$X_{A_i/\mathfrak m_i^{n_i}}$ (again because the colimit category
is invariant under universal homeomorphisms, see above).
Then Lemma \ref{lemma-trivial-fibres-dvr} tells us that
$\mathcal{E}$ is free on $X_{A_i}$.
Thus finally we conclude by applying Lemma \ref{lemma-trivial-over-dvrs}.

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