The Stacks project

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 37.72.3. 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. $\square$


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