# stacks/stacks-project

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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