# stacks/stacks-project

Add in all the missing details from 9a7776c

 @@ -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}.