The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 34.8.13. Let $S$ be a scheme. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. The functors

\[ \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O}) \quad \text{and}\quad \mathit{QCoh}(\mathcal{O}_ S) \longrightarrow \textit{Mod}(S_\tau , \mathcal{O}) \]

defined by the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ seen in Proposition 34.8.11 are

  1. fully faithful,

  2. compatible with direct sums,

  3. compatible with colimits,

  4. right exact,

  5. exact as a functor $\mathit{QCoh}(\mathcal{O}_ S) \to \textit{Mod}(S_{\acute{e}tale}, \mathcal{O})$,

  6. not exact as a functor $\mathit{QCoh}(\mathcal{O}_ S) \to \textit{Mod}((\mathit{Sch}/S)_\tau , \mathcal{O})$ in general,

  7. given two quasi-coherent $\mathcal{O}_ S$-modules $\mathcal{F}$, $\mathcal{G}$ we have $(\mathcal{F} \otimes _{\mathcal{O}_ S} \mathcal{G})^ a = \mathcal{F}^ a \otimes _\mathcal {O} \mathcal{G}^ a$,

  8. given two quasi-coherent $\mathcal{O}_ S$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation we have $(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ S}(\mathcal{F}, \mathcal{G}))^ a = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}^ a, \mathcal{G}^ a)$, and

  9. given a short exact sequence $0 \to \mathcal{F}_1^ a \to \mathcal{E} \to \mathcal{F}_2^ a \to 0$ of $\mathcal{O}$-modules then $\mathcal{E}$ is quasi-coherent1, i.e., $\mathcal{E}$ is in the essential image of the functor.

Proof. Part (1) we saw in Proposition 34.8.11.

We have seen in Schemes, Section 25.24 that a colimit of quasi-coherent sheaves on a scheme is a quasi-coherent sheaf. Moreover, in Remark 34.8.6 we saw that $\mathcal{F} \mapsto \mathcal{F}^ a$ is the pullback functor for a morphism of ringed sites, hence commutes with all colimits, see Modules on Sites, Lemma 18.14.3. Thus (3) and its special case (3) hold.

This also shows that the functor is right exact (i.e., commutes with finite colimits), hence (4).

The functor $\mathit{QCoh}(\mathcal{O}_ S) \to \mathit{QCoh}(S_{\acute{e}tale}, \mathcal{O})$, $\mathcal{F} \mapsto \mathcal{F}^ a$ is left exact because an ├ętale morphism is flat, see Morphisms, Lemma 28.34.12. This proves (5).

To see (6), suppose that $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$. Then $2 : \mathcal{O}_ S \to \mathcal{O}_ S$ is injective but the associated map of $\mathcal{O}$-modules on $(\mathit{Sch}/S)_\tau $ isn't injective because $2 : \mathbf{F}_2 \to \mathbf{F}_2$ isn't injective and $\mathop{\mathrm{Spec}}(\mathbf{F}_2)$ is an object of $(\mathit{Sch}/S)_\tau $.

We omit the proofs of (7) and (8).

Let $0 \to \mathcal{F}_1^ a \to \mathcal{E} \to \mathcal{F}_2^ a \to 0$ be a short exact sequence of $\mathcal{O}$-modules with $\mathcal{F}_1$ and $\mathcal{F}_2$ quasi-coherent on $S$. Consider the restriction

\[ 0 \to \mathcal{F}_1 \to \mathcal{E}|_{S_{Zar}} \to \mathcal{F}_2 \]

to $S_{Zar}$. By Proposition 34.8.10 we see that on any affine $U \subset S$ we have $H^1(U, \mathcal{F}_1^ a) = H^1(U, \mathcal{F}_1) = 0$. Hence the sequence above is also exact on the right. By Schemes, Section 25.24 we conclude that $\mathcal{F} = \mathcal{E}|_{S_{Zar}}$ is quasi-coherent. Thus we obtain a commutative diagram

\[ \xymatrix{ & \mathcal{F}_1^ a \ar[r] \ar[d] & \mathcal{F}^ a \ar[r] \ar[d] & \mathcal{F}_2^ a \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{F}_1^ a \ar[r] & \mathcal{E} \ar[r] & \mathcal{F}_2^ a \ar[r] & 0 } \]

To finish the proof it suffices to show that the top row is also right exact. To do this, denote once more $U = \mathop{\mathrm{Spec}}(A) \subset S$ an affine open of $S$. We have seen above that $0 \to \mathcal{F}_1(U) \to \mathcal{E}(U) \to \mathcal{F}_2(U) \to 0$ is exact. For any affine scheme $V/U$, $V = \mathop{\mathrm{Spec}}(B)$ the map $\mathcal{F}_1^ a(V) \to \mathcal{E}(V)$ is injective. We have $\mathcal{F}_1^ a(V) = \mathcal{F}_1(U) \otimes _ A B$ by definition. The injection $\mathcal{F}_1^ a(V) \to \mathcal{E}(V)$ factors as

\[ \mathcal{F}_1(U) \otimes _ A B \to \mathcal{E}(U) \otimes _ A B \to \mathcal{E}(U) \]

Considering $A$-algebras $B$ of the form $B = A \oplus M$ we see that $\mathcal{F}_1(U) \to \mathcal{E}(U)$ is universally injective (see Algebra, Definition 10.81.1). Since $\mathcal{E}(U) = \mathcal{F}(U)$ we conclude that $\mathcal{F}_1 \to \mathcal{F}$ remains injective after any base change, or equivalently that $\mathcal{F}_1^ a \to \mathcal{F}^ a$ is injective. $\square$

[1] Warning: This is misleading. See part (6).

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