The Stacks project

38.39 Descent vector bundles in positive characteristic

A reference for this section is [Witt-Grass].

For a scheme $S$ let us denote $\textit{Vect}(S)$ the category of finite locally free $\mathcal{O}_ S$-modules. Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_ p$. In this section we will work with the category

\[ \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(S) = \mathop{\mathrm{colim}}\nolimits \left( \textit{Vect}(S) \xrightarrow {F^*} \textit{Vect}(S) \xrightarrow {F^*} \textit{Vect}(S) \xrightarrow {F^*} \ldots \right) \]

where $F : S \to S$ is the absolute Frobenius morphism. In down to earth terms an object of this category is a pair $(\mathcal{E}, n)$ where $\mathcal{E}$ is a finite locally free $\mathcal{O}_ S$-module and $n \geq 0$ is an integer. For morphisms we take

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(S)}((\mathcal{E}, n), (\mathcal{G}, m)) = \mathop{\mathrm{colim}}\nolimits _ N \mathop{\mathrm{Hom}}\nolimits _ S(F^{N - n, *}\mathcal{E}, F^{N - m, *}\mathcal{G}) \]

where $F : S \to S$ is the absolute Frobenius morphism of $S$. Thus the object $(\mathcal{E}, n)$ is isomorphic to the object $(F^*\mathcal{E}, n + 1)$.

Lemma 38.39.1. Let $p$ be a prime number. Let $S$ be a quasi-compact and quasi-separated scheme over $\mathbf{F}_ p$. The category $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(S)$ is equivalent to the category of finite locally free modules over the sheaf of rings $\mathop{\mathrm{colim}}\nolimits _ F \mathcal{O}_ S$ on $S$.

Proof. Omitted. $\square$

Lemma 38.39.2. Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example 38.37.10. Then the functor

\[ \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Z) \times _{\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(E)} \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X') \]

is an equivalence.

Proof. Let $A, f, J$ be as in Example 38.37.10. Since all our schemes are affine and since we have internal Hom in the category of vector bundles, the fully faithfulness of the functor follows if we can show that

\[ \mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A = \mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/J \times _{\mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/fA + J} \mathop{\mathrm{colim}}\nolimits P \otimes _{A, F^ N} A/fA \]

for a finite projective $A$-module $P$. After writing $P$ as a summand of a finite free module, this follows from the case where $P$ is finite free. This case immediately reduces to the case $P = A$. The case $P = A$ follows from Lemma 38.38.2 (in fact we proved this case directly in the proof of this lemma).

Essential surjectivity. Here we obtain the following algebra problem. Suppose $P_1$ is a finite projective $A/J$-module, $P_2$ is a finite projective $A/fA$-module, and

\[ \varphi : P_1 \otimes _{A/J} A/fA + J \longrightarrow P_2 \otimes _{A/fA} A/fA + J \]

is an isomorphism. Goal: show that there exists an $N$, a finite projective $A$-module $P$, an isomorphism $\varphi _1 : P \otimes _ A A/J \to P_1 \otimes _{A/J, F^ N} A/J$, and an isomorphism $\varphi _2 : P \otimes _ A A/fA \to P_2 \otimes _{A/fA, F^ N} A/fA$ compatible with $\varphi $ in an obvious manner. This can be seen as follows. First, observe that

\[ A/(J \cap fA) = A/J \times _{A/fA + J} A/fA \]

Hence by More on Algebra, Lemma 15.6.9 there is a finite projective module $P'$ over $A/(J \cap fA)$ which comes with isomorphisms $\varphi '_1 : P' \otimes _ A A/J \to P_1$ and $\varphi _2 : P' \otimes _ A A/fA \to P_2$ compatible with $\varphi $. Since $J$ is a finitely generated ideal and $f$-power torsion we see that $J \cap fA$ is a nilpotent ideal. Hence for some $N$ there is a factorization

\[ A \xrightarrow {\alpha } A/(J \cap fA) \xrightarrow {\beta } A \]

of $F^ N$. Setting $P = P' \otimes _{A/(J \cap fA), \beta } A$ we conclude. $\square$

Lemma 38.39.3. Let $p$ be a prime number. Consider an almost blowup square $X, X', Z, E$ in characteristic $p$ as in Example 38.37.11. Then the functor

\[ G : \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Z) \times _{\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(E)} \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X') \]

is an equivalence.

Proof. Fully faithfulness. Suppose that $(\mathcal{E}, n)$ and $(\mathcal{F}, m)$ are objects of $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X)$. Let $(a, b) : G(\mathcal{E}, n) \to G(\mathcal{F}, m)$ be a morphism in the RHS. We may choose $N \gg 0$ and think of $a$ as a map $a : F^{N - n, *}\mathcal{E}|_ Z \to F^{N - m, *}\mathcal{F}|_ Z$ and $b$ as a map $b : F^{N - n, *}\mathcal{E}|_{X'} \to F^{N - m, *}\mathcal{F}|_{X'}$ agreeing over $E$. Choose a finite affine open covering $X = X_1 \cup \ldots \cup X_ n$ such that $\mathcal{E}|_{X_ i}$ and $\mathcal{F}|_{X_ i}$ are finite free $\mathcal{O}_{X_ i}$-modules. For each $i$ the base change

\[ \xymatrix{ E_ i \ar[r] \ar[d] & X'_ i \ar[d] \\ Z_ i \ar[r] & X_ i } \]

is another almost blow up square as in Example 38.37.11. For these squares we know that

\[ \mathop{\mathrm{colim}}\nolimits _ F H^0(X_ i, \mathcal{O}_{X_ i}) = \mathop{\mathrm{colim}}\nolimits _ F H^0(Z_ i, \mathcal{O}_{Z_ i}) \times _{\mathop{\mathrm{colim}}\nolimits _ F H^0(E_ i, \mathcal{O}_{E_ i})} \mathop{\mathrm{colim}}\nolimits _ F H^0(X'_ i, \mathcal{O}_{X'_ i}) \]

by Lemma 38.38.2 (see proof of the lemma). Hence after increasing $N$ we may assume the maps $a|_{Z_ i}$ and $b|_{X'_ i}$ come from maps $c_ i : F^{N - n, *}\mathcal{E}|_{X_ i} \to F^{N - m, *}\mathcal{F}|_{X_ i}$. After possibly increasing $N$ we may assume $c_ i$ and $c_ j$ agree on $X_ i \cap X_ j$. Thus these maps glue to give the desired morphism $(\mathcal{E}, n) \to (\mathcal{F}, m)$ in the LHS.

Essential surjectivity. Let $(\mathcal{F}, \mathcal{G}, \varphi )$ be a triple consisting of a finite locally free $\mathcal{O}_ Z$-module $\mathcal{F}$, a finite locally free $\mathcal{O}_{X'}$-module $\mathcal{G}$, and an isomorphism $\varphi : \mathcal{F}|_ E \to \mathcal{G}|_ E$. We have to show that after replacing this triple by a Frobenius power pullback, it comes from a finite locally free $\mathcal{O}_ X$-module.

Noetherian reduction; we urge the reader to skip this paragraph. Recall that $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/(f_1, f_2))$, $X' = \text{Proj}(A[T_0, T_1]/(f_2T_0 - f_1T_1))$, and $E = \mathbf{P}^1_ Z$. By Limits, Lemma 32.10.3 we can find a finitely generated $\mathbf{F}_ p$-subalgebra $A_0 \subset A$ containing $f_1$ and $f_2$ such that the triple $(\mathcal{F}, \mathcal{G}, \varphi )$ descends to $X_0 = \mathop{\mathrm{Spec}}(A_0)$ and $Z_0 = \mathop{\mathrm{Spec}}(A_0/(f_1, f_2))$, $X_0' = \text{Proj}(A_0[T_0, T_1]/(f_2T_0 - f_1T_1))$, and $E_0 = \mathbf{P}^1_{Z_0}$. Thus we may assume our schemes are Noetherian.

Assume $X$ is Noetherian. We may choose a finite affine open covering $X = X_1 \cup \ldots \cup X_ n$ such that $\mathcal{F}|_{Z \cap X_ i}$ is free. Since we can glue objects of $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X)$ in the Zariski topology (Lemma 38.39.1), and since we already know fully faithfulness over $X_ i$ and $X_ i \cap X_ j$ (see first paragraph of the proof), it suffices to prove the existence over each $X_ i$. This reduces us to the case discussed in the next paragraph.

Assume $X$ is Noetherian and $\mathcal{F} = \mathcal{O}_ Z^{\oplus r}$. Using $\varphi $ we get an isomorphism $\mathcal{O}_ E^{\oplus r} \to \mathcal{G}|_ E$. Let $I = (f_1, f_2) \subset A$. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the ideal sheaf of $E$; it is globally generated by $f_1$ and $f_2$. For any $n$ there is a surjection

\[ (\mathcal{I}^ n/\mathcal{I}^{n + 1})^{\oplus r} = \mathcal{I}^ n/\mathcal{I}^{n + 1} \otimes _{\mathcal{O}_ E} \mathcal{G}|_ E \longrightarrow \mathcal{I}^ n\mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} \]

Hence the first cohomology group of this module is zero. Here we use that $E = \mathbf{P}^1_ Z$ and hence its structure sheaf and in fact any globally generated quasi-coherent module has vanishing $H^1$. Compare with More on Morphisms, Lemma 37.66.3. Then using the short exact sequences

\[ 0 \to \mathcal{I}^ n\mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} \to \mathcal{G}/\mathcal{I}^{n + 1}\mathcal{G} \to \mathcal{G}/\mathcal{I}^ n\mathcal{G} \to 0 \]

and induction, we see that

\[ \mathop{\mathrm{lim}}\nolimits H^0(X', \mathcal{G}/\mathcal{I}^ n\mathcal{G}) \to H^0(E, \mathcal{G}|_ E) = H^0(E, \mathcal{O}_ E^{\oplus r}) = A/I^{\oplus r} \]

is surjective. By the theorem on formal functions (Cohomology of Schemes, Theorem 30.20.5) this implies that

\[ H^0(X', \mathcal{G}) \to H^0(E, \mathcal{G}|_ E) = H^0(E, \mathcal{O}_ E^{\oplus r}) = A/I^{\oplus r} \]

is surjective. Thus we can choose a map $\alpha : \mathcal{O}_{X'}^{\oplus r} \to \mathcal{G}$ which is compatible with the given trivialization of $\mathcal{G}|_ E$. Thus $\alpha $ is an isomorphism over an open neighbourhood of $E$ in $X'$. Thus every point of $Z$ has an affine open neighbourhood where we can solve the problem. Since $X' \setminus E \to X \setminus Z$ is an isomorphism, the same holds for points of $X$ not in $Z$. Thus another Zariski glueing argument finishes the proof. $\square$

Proposition 38.39.4. Let $p$ be a prime number. Let $S$ be a scheme in characteristic $p$. Then the category fibred in groupoids

\[ p : \mathcal{S} \longrightarrow (\mathit{Sch}/S)_ h \]

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

\[ \textit{Vect}(Y) \longrightarrow \textit{Vect}(X) \times _{\textit{Vect}(X \times _ Y X)} \textit{Vect}(X) \]

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. $\square$

Lemma 38.39.5. 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.

Proof. We first reduce to the case where $X \to S$ has a section. Say $S = \mathop{\mathrm{Spec}}(A)$. 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 10.120.18) and the fact that $\kappa (\xi )$ is a finite extension of the fraction field of $A$. By the valuative criterion of properness (Morphisms, Lemma 29.42.1) we get a $B$-valued point $\tau : \mathop{\mathrm{Spec}}(B) \to X$ which induces a section $\sigma : \mathop{\mathrm{Spec}}(B) \to X_ B$. 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 30.5.2) 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 $g \in \Gamma (X_\eta , \mathcal{O}_{X_ B})$ which 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 33.9.3). Thus it suffices to prove the lemma for the base change $X_ B \to \mathop{\mathrm{Spec}}(B)$.

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 (and only if) the induced map $H^0(K) = H^0(X, \mathcal{E}) \to A^{\oplus r}$ is surjective.

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. Let $\pi \in A$ be a uniformizer. Observe that

\[ K \otimes _ A^\mathbf {L} A/\pi ^ m A = R\Gamma (X, \mathcal{E} \xrightarrow {\pi ^ m} \mathcal{E}) \]

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 30.10.3). 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

\[ K \cong \bigoplus H^ i(K)[-i] \quad \text{and}\quad L \cong \bigoplus H^ i(L)[-i] \]

in $D(A)$. See More on Algebra, Example 15.68.3. 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 30.19.2. By More on Algebra, Lemma 15.121.3 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 for $K$ and the isomorphism $K \otimes _ A^\mathbf {L} A/\pi ^ m A \cong L \otimes _ A^\mathbf {L} A/\pi ^ m A$ it 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

\[ H^0(K \otimes ^\mathbf {L}_ A A/\pi ^{e + 1} A) = H^0(K)/\pi ^{e + 1}H^0(K) \oplus H^1(K)[\pi ^{e + 1}] \longrightarrow (A/\pi ^{e + 1} A)^{\oplus r} \]

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 $X$.) $\square$

Lemma 38.39.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Assume

  1. $f$ is flat and proper and $\mathcal{O}_ S = f_*\mathcal{O}_ X$,

  2. $S$ is a normal Noetherian scheme,

  3. the pullback of $\mathcal{E}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is free for every codimension $1$ point $s \in S$.

Then $\mathcal{E}$ is isomorphic to the pullback of a finite locally free $\mathcal{O}_ S$-module.

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 30.5.2) and assumptions (1) and (3) we see that the pullback of this to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is an isomorphism for every codimension $1$ point $s \in S$. By Divisors, Lemma 31.2.11 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 10.163.2. Since $S$ is a normal Noetherian scheme and $\dim (\mathcal{O}_{S, s}) \geq 2$ we have $\text{depth}(\mathcal{O}_{S, s}) \geq 2$, see Properties, Lemma 28.12.5. Thus we get what we want from Divisors, Lemma 31.6.6. $\square$

We can use the results above to prove the following miraculous statement.

Theorem 38.39.7. 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

\[ \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Y) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X) \]

is fully faithful with essential image described as follows. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Assume for all $y \in Y$ there exists integers $n_ y, r_ y \geq 0$ such that

\[ F^{n_ y, *}\mathcal{E}|_{X_{y, red}} \cong \mathcal{O}_{X_{y, red}}^{\oplus r_ y} \]

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.

Proof. Proof of fully faithfulness. Since vectorbundles on $Y$ are locally trivial, this reduces to the statement that

\[ \mathop{\mathrm{colim}}\nolimits _ F \Gamma (Y, \mathcal{O}_ Y) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \Gamma (X, \mathcal{O}_ X) \]

is bijective. Since $\{ X \to Y\} $ is an h covering, this will follow from Lemma 38.38.2 if we can show that the two maps

\[ \mathop{\mathrm{colim}}\nolimits _ F \Gamma (X, \mathcal{O}_ X) \longrightarrow \mathop{\mathrm{colim}}\nolimits _ F \Gamma (X \times _ Y X, \mathcal{O}_{X \times _ Y X}) \]

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 $\kappa (y)$, see Varieties, Lemma 33.9.3. 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.

Description of essential image. Let $\mathcal{E}$ be as in the statement of the proposition. We first reduce to the Noetherian case.

Let $y \in Y$ be a point and view it as a morphism $y \to Y$ 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 32.7.1 and 32.4.13.) For each $i$ set $Z_ i = Y_ i \times _ Y X$. Then $X_ y = \mathop{\mathrm{lim}}\nolimits Z_ i$ and $X_{y, red} = \mathop{\mathrm{lim}}\nolimits Z_{i, red}$. By Limits, Lemma 32.10.2 we can find an $i$ such that $F^{n_ y, *}\mathcal{E}|_{Z_{i, red}} \cong \mathcal{O}_{Z_{i, red}}^{\oplus r_ y}$. Fix $i$. We have $Z_{i, red} = \mathop{\mathrm{lim}}\nolimits Z_{i, j}$ where $Z_{i, j} \to Z_ i$ is a thickening of finite presentation (Limits, Lemma 32.9.4). Using the same lemma as before we can find a $j$ such that $F^{n_ y, *}\mathcal{E}|_{Z_{i, j}} \cong \mathcal{O}_{Z_{i, j}}^{\oplus r_ y}$. 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 $F^{n_ y, *}\mathcal{E}|_{Z_ y} \cong \mathcal{O}_{Z_ y}^{\oplus r_ y}$. Observe that the image of $Y_ y \to Y$ is constructible (Morphisms, Lemma 29.22.2). Since $Y$ is quasi-compact in the constructible topology (Topology, Lemma 5.23.2 and Properties, Lemma 28.2.4) we conclude that there are a finite number of morphisms

\[ Y_1 \to Y,\ Y_2 \to Y,\ \ldots ,\ Y_ N \to Y \]

of finite presentation such that $Y = \bigcup \mathop{\mathrm{Im}}(Y_ a \to Y)$ set theoretically and such that for each $a \in \{ 1, \ldots , N\} $ 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 $F^{n_ a, *}\mathcal{E}|_{Z_ a} \cong \mathcal{O}_{Z_ a}^{\oplus r_ a}$.

Formulated in this way, the condition descends to an absolute Noetherian approximation. We stronly urge the reader to skip this paragraph. First write $Y = \mathop{\mathrm{lim}}\nolimits _{i \in I} Y_ i$ as a cofiltered limit of schemes of finite type over $\mathbf{F}_ p$ with affine transition morphisms (Limits, Lemma 32.7.1). 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 32.10.1. 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 32.10.3. 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 37.26.6. Then $Y \to Y_0$ maps into $E$, see More on Morphisms, Lemma 37.26.2. Thus $Y_ i \to Y_0$ maps into $E$ for $i \gg 0$, see Limits, Lemma 32.4.10. Hence we see that the fibres of $f_ i$ are geometrically connected for $i \gg 0$. By Limits, Lemma 32.10.1 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 32.8.5 and 32.8.14). Since $Z_ a = \mathop{\mathrm{lim}}\nolimits Z_{i, a}$ we find that after increasing $i$ we may assume $F^{n_ a, *}\mathcal{E}_ i|_{Z_{i, a}} \cong \mathcal{O}_{Z_{i, a}}^{\oplus r_ a}$, see Limits, Lemma 32.10.2. Finally, after increasing $i$ one more time we may assume $\coprod Y_{i, a} \to Y_ i$ is surjective by Limits, Lemma 32.8.14. 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 $\mathcal{E}_ i$.

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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Y)$ which pulls back to the object of $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(X)$ determined by $\mathcal{E}$. By the fully faithfulness already proven and because of Proposition 38.39.4 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. 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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(T) = \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(T')$ 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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(T) = \mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(T')$ as $\{ T \to T'\} $ is a h covering such that the diagonal $T \subset T \times _{T'} T$ is a thickening.

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 37.49.4. Then $Y' \to Y$ is a universal homeomorphism 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.

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 29.27.2. By Lemma 38.31.1 there is a $V$-admissible blowing up $Y' \to Y$ such that the strict transform $X'$ of $X$ is flat over $Y'$. Observe that $\dim (Y') = \dim (Y)$ as $Y$ and $Y'$ have a common dense open subscheme. By More on Morphisms, Lemma 37.49.7 and the fact that $V \subset Y'$ is dense all fibres of $f' : X' \to Y'$ are geometrically connected. We still have $(f'_*\mathcal{O}_{X'})|_ V = \mathcal{O}_ V$. 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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(Y')$ 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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(E)$ 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

\[ \{ E \times _ Y X \to Y' \times _ Y X, X' \to Y' \times _ Y X\} \]

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.

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 29.54). This is a finite surjective morphism (Morphisms, Lemma 29.54.10 and 29.18.2) 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).

Assume $Y$ is a normal Noetherian scheme, that $f$ is flat, and 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 $\mathop{\mathrm{colim}}\nolimits _ F \textit{Vect}(-)$ 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 $z_1, \ldots , z_ t \in Z$ 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 38.39.5 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 $X_{A_ i/\mathfrak m_ i^{n_ i}}$ (again because the colimit category is invariant under universal homeomorphisms, see above). Then Lemma 38.39.5 tells us that $\mathcal{E}$ is free on $X_{A_ i}$. Thus finally we conclude by applying Lemma 38.39.6. $\square$


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