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Rough draft argument descent vbs char p

Thanks to Bhargav Bhatt
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aisejohan committed Sep 10, 2018
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@@ -11721,6 +11721,355 @@ \section{Descent vector bundles in positive characteristic}
Lemmas \ref{lemma-vector-bundle-I} and \ref{lemma-vector-bundle-II}.
\end{proof}
\begin{lemma}
\label{lemma-trivial-fibres-dvr}
Let $f : X \to S$ be a proper morphism with geometrically connected fibres
where $S$ is the spectrum of a discrete valuation ring. Denote $\eta \in S$
the generic point and denote $X_n \subset X$ the closed subscheme
cutout by the $n$th power of a uniformizer on $S$.
Then there exists
an integer $n$ such that the following is true: any finite
locally free $\mathcal{O}_X$-module $\mathcal{E}$
such that $\mathcal{E}|_{X_\eta}$ and $\mathcal{E}|_{X_n}$
are free, is free.
\end{lemma}
\begin{proof}
We first reduce to the case where $X \to S$ has a section. Say $S = \Spec(A)$.
Choose a closed point $\xi$ of $X_\eta$. Choose an extension
of discrete valuation rings $A \subset B$ such that the fraction field
of $B$ is $\kappa(\xi)$. This is possible by Krull-Akizuki
(Algebra, Lemma \ref{algebra-lemma-integral-closure-Dedekind})
and the fact that $\kappa(\xi)$ is a finite extension of the
fraction field of $A$.
By the valuative criterion of properness
(Morphisms, Lemma \ref{morphisms-lemma-characterize-proper})
we get a $B$-valued point $\tau : \Spec(B) \to X$
which induces a section $\sigma : \Spec(B) \to X_B$.
For a finite locally free $\mathcal{O}_X$-module $\mathcal{E}$
let $\mathcal{E}_B$ be the pullback to the base change $X_B$.
By flat base change
(Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology})
we see that $H^0(X_B, \mathcal{E}_B) = H^0(X, \mathcal{E}) \otimes_A B$.
Thus if $\mathcal{E}_B$ is free of rank $r$, then the sections in
$H^0(X, \mathcal{E})$ generate the free $B$-module
$\tau^*\mathcal{E} = \sigma^*\mathcal{E}_B$.
In particular, we can find $r$ global sections $s_1, \ldots, s_r$
of $\mathcal{E}$ which generate $\tau^*\mathcal{E}$. Then
$$
s_1, \ldots, s_r :
\mathcal{O}_X^{\oplus r}
\longrightarrow
\mathcal{E}
$$
is a map of finite locally free $\mathcal{O}_X$-modules of rank $r$
and the pullback to $X_B$ is a map of free $\mathcal{O}_{X_B}$-modules
which restricts to an isomorphism in one point of each fibre.
Taking the determinant we get a function
$g \in \Gamma(X_\eta, \mathcal{O}_{X_B})$
wich is invertible in one point of each fibre.
As the fibres are proper and connected, we see that $g$
must be invertible (details omitted; hint: use Varieties, Lemma
\ref{varieties-lemma-proper-geometrically-reduced-global-sections}).
Thus it suffices to prove the lemma for the base change $X_B \to \Spec(B)$.
\medskip\noindent
Assume we have a section $\sigma : S \to X$. Let $\mathcal{E}$
be a finite locally free $\mathcal{O}_X$-module which is assumed
free on the generic fibre and on $X_n$ (we will choose $n$ later).
Choose an isomorphism $\sigma^*\mathcal{E} = \mathcal{O}_S^{\oplus r}$.
Consider the map
$$
K = R\Gamma(X, \mathcal{E}) \longrightarrow
R\Gamma(S, \sigma^*\mathcal{E}) = A^{\oplus r}
$$
in $D(A)$. Arguing as above, we see $\mathcal{E}$ is free if
the induced map $H^0(X, \mathcal{E}) \to A^{\oplus r}$ is surjective.
\medskip\noindent
Set $L = R\Gamma(X, \mathcal{O}_X^{\oplus r})$ and observe that the
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
$$
K \otimes_A^\mathbf{L} A/\pi^m A =
R\Gamma(X, \mathcal{E} \xrightarrow{\pi^m} \mathcal{E})
$$
and similarly for $L$.
Denote $\mathcal{E}_{tors} \subset \mathcal{E}$ the coherent subsheaf of
sections supported on the special fibre and similarly for other
$\mathcal{O}_X$-modules. Choose $k > 0$ such that
$(\mathcal{O}_X)_{tors} \to \mathcal{O}_X/\pi^k \mathcal{O}_X$
is injective (Cohomology of Schemes, Lemma \ref{coherent-lemma-Artin-Rees}).
Since $\mathcal{E}$ is locally free, we see
that $\mathcal{E}_{tors} \subset \mathcal{E}/\pi^k\mathcal{E}$.
Then for $n \geq m + k$ we have isomorphisms
\begin{align*}
(\mathcal{E} \xrightarrow{\pi^m} \mathcal{E})
& \cong
(\mathcal{E}/\pi^k\mathcal{E} \xrightarrow{\pi^m}
\mathcal{E}/\pi^{k + m}\mathcal{E}) \\
& \cong
(\mathcal{O}_X^{\oplus r}/\pi^k\mathcal{O}_X^{\oplus r} \xrightarrow{\pi^m}
\mathcal{O}_X^{\oplus r}/\pi^{k + m}\mathcal{O}_X^{\oplus r}) \\
& \cong
(\mathcal{O}_X^{\oplus r} \xrightarrow{\pi^m} \mathcal{O}_X^{\oplus r})
\end{align*}
in $D(\mathcal{O}_X)$. This determines an isomorphism
$$
K \otimes_A^\mathbf{L} A/\pi^m A \cong L \otimes_A^\mathbf{L} A/\pi^m A
$$
in $D(A)$ (holds when $n \geq m + k$). Observe that these isomorphisms
are compatible with pulling back by $\sigma$ hence in particular
we conclude that
$K \otimes_A^\mathbf{L} A/\pi^m A \to (A/\pi^m A)^{\oplus r}$
defines an surjection on degree $0$ cohomology modules (as
this is true for $L$).
Since $A$ is a discrete valuation ring, we have
$$
K \cong \bigoplus H^i(K)[-i]
\quad\text{and}\quad
L \cong
\bigoplus H^i(L)^{\oplus r}[-i]
$$
in $D(A)$. See More on Algebra, Example
\ref{more-algebra-example-finite-injective-finite-global-dimension}.
The cohomology groups $H^i(K) = H^i(X, \mathcal{E})$ and
$H^i(L) = H^i(X, \mathcal{O}_X)^{\oplus r}$
are finite $A$-modules by Cohomology of Schemes, Lemma
\ref{coherent-lemma-proper-over-affine-cohomology-finite}.
By More on Algebra, Lemma
\ref{more-algebra-lemma-generalized-valuation-ring-modules}
these modules are direct sums of cyclic modules.
We have seen above that the rank $\beta_i$ of the
free part of $H^i(K)$ and $H^i(L)$ are the same.
Next, observe that
$$
H^i(L \otimes_A^\mathbf{L} A/\pi^m A) =
H^i(L)/\pi^m H^i(L) \oplus H^{i + 1}(L)[\pi^m]
$$
and similarly for $K$. Let $e$ be the largest integer
such that $A/\pi^eA$ occurs as a summand of $H^i(X, \mathcal{O}_X)$,
or equivalently $H^i(L)$, for some $i$. Then taking $m = e + 1$
we see that $H^i(L \otimes_A^\mathbf{L} A/\pi^m A)$ is a direct sum of
$\beta_i$ copies of $A/\pi^m A$ and some other cyclic modules
each annihilated by $\pi^e$. By the same reasoning
$H^i(L \otimes_A^\mathbf{L} A/\pi^m A)$ is a direct sum of
$\beta_i$ copies of $A/\pi^m A$ and some other cyclic modules.
It immediately follows that $H^i(K)$
cannot have any cyclic summands of the form $A/\pi^l A$
with $l > e$. (It also follows that $K$ is isomorphic to $L$
as an object of $D(A)$, but we won't need this.)
Then the only way the map
$$
H^0(K \otimes^\mathbf{L}_A A/\pi^{e + 1} A) =
H^0(K)/\pi^{e + 1}H^0(K) \oplus H^1(K)[\pi^{e + 1}]
\longrightarrow
A/\pi^{e + 1} A
$$
is surjective, is if it is surjective on the
first summand. This is what we wanted to show.
(To be precise, the integer $n$ in the statement of
the lemma, if there is a section $\sigma$,
should be equal to $k + e + 1$ where $k$ and $e$ are as above
and depend only on $X$.)
\end{proof}
\begin{lemma}
\label{lemma-trivial-over-dvrs}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module.
Assume
\begin{enumerate}
\item $f$ is flat and proper and $\mathcal{O}_S = f_*\mathcal{O}_X$,
\item $S$ is a normal Noetherian scheme,
\item the pullback of $\mathcal{E}$ to $X \times_S \Spec(\mathcal{O}_{S, s})$
is free for every codimension $1$ point $s \in S$.
\end{enumerate}
Then $\mathcal{E}$ is isomorphic to the pullback of a finite
locally free $\mathcal{O}_S$-module.
\end{lemma}
\begin{proof}
We will prove the canonical map
$$
\Phi : f^*f_*\mathcal{E} \longrightarrow \mathcal{E}
$$
is an isomorphism. By flat base change (Cohomology of Schemes, Lemma
\ref{coherent-lemma-flat-base-change-cohomology})
and assumptions (1) and (3) we see that
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 $\text{depth} (f^*f_*\mathcal{E})_x \geq 2$
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
Algebra, Lemma \ref{algebra-lemma-apply-grothendieck-module}.
Since $S$ is a normal Noetherian scheme
and $\dim(\mathcal{O}_{S, s}) \geq 2$
we can choose a regular sequence $a, b \in \mathfrak m_s$, see
Properties, Lemma \ref{properties-lemma-criterion-normal}.
We will show that $a, b$ is a regular sequence on
$(f_*\mathcal{E})_s$. To do this we
may replace $S$ by the spectrum of $\mathcal{O}_{S, s}$;
details omitted. Since $f$ is flat and $\mathcal{E}$ locally free,
we have a short exact sequence
$$
0 \to \mathcal{E} \xrightarrow{a} \mathcal{E}
\to \mathcal{E}/a\mathcal{E} \to 0
$$
Hence $a : f_*\mathcal{E} \to f_*\mathcal{E}$ is an injective
map whose cokernel is a submodule of
$f_*(\mathcal{E}/a\mathcal{E})$. Again using that $f$ is flat
and $\mathcal{E}$ locally free we find that
$$
b : \mathcal{E}/a\mathcal{E} \to \mathcal{E}/a\mathcal{E}
$$
is injective. Hence $b$ is injective on
$f_*(\mathcal{E}/a\mathcal{E})$ and hence
on $\mathcal{E}/a\mathcal{E}$ as desired.
\end{proof}
\noindent
We can use the results above to prove the following
miraculous statement.
\begin{theorem}
\label{theorem-pullback-trivial-fibres}
Let $p$ be a prime number. Let $Y$ be a quasi-compact and quasi-separated
scheme over $\mathbf{F}_p$.
Let $f : X \to Y$ be a proper, surjective morphism of finite presentation
with geometrically connected fibres.
Then the functor
$$
\colim_F \textit{Vect}(Y) \longrightarrow \colim_F \textit{Vect}(X)
$$
is fully faithful with essential image described as follows.
Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_X$-module.
Assume
$$
\mathcal{E}|_{X_{y, red}} \cong \mathcal{O}_{X_{y, red}}^{\oplus r_y}
$$
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.
\end{theorem}
\begin{proof}
Proof of fully faithfulness. Since vectorbundles on $Y$ are locally
trivial, this reduces to the statement that
$$
\colim_F \Gamma(Y, \mathcal{O}_Y)
\longrightarrow
\colim_F \Gamma(X, \mathcal{O}_X)
$$
is bijective. Since $\{X \to Y\}$ is an h covering, this will
follow from Lemma \ref{lemma-h-sheaf-colim-F} if we can show that the two maps
$$
\colim_F \Gamma(X, \mathcal{O}_X)
\longrightarrow
\colim_F \Gamma(X \times_Y X, \mathcal{O}_{X \times_Y X})
$$
are equal. Let $g \in \Gamma(X, \mathcal{O}_X)$
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 $\kappa(y)$.
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
element of the residue field at any point of
$(X \times_Y X)_y = X_y \times_y X_y$.
Hence $g_1 - g_2$ restricts to zero on $(X \times_Y X)_{red}$.
Hence $(g_1 - g_2)^n = 0$ for some $n$ which we may take
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.
By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation}
we can find a pair $(i, j)$ such that
$\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_y}$.
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
$$
Y_1 \to Y, Y_2 \to Y, \ldots, Y_n \to Y
$$
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
$\mathcal{E}|_{Z_i} \cong \mathcal{O}_{Z_i}^{\oplus r_i}$.
\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
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 $X \to Y' \to Y$.
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
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
\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
$$
g : T = (Y')^\nu \times_{Y'} X' \longrightarrow (Y')^\nu = S
$$
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.
\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.
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 $z_1, \ldots, z_t \in Z$
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).
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}.
\end{proof}

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