The Stacks project

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

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

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 32.7.2 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

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 32.7.2). 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 37.28.6. Then $Y \to Y_0$ maps into $E$, see More on Morphisms, Lemma 37.28.2. 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 32.8.15). 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 32.8.15. 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 37.53.4. 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 37.53.7 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

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

is a h covering given by a pair of closed immersions with

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. $\square$