The Stacks project

Special case of [Theorem 1.2, Brandenburg]

Proposition 56.6.6. Let $X$ and $Y$ be quasi-compact and quasi-separated schemes. If $F : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ is an equivalence, then there exists an isomorphism $f : Y \to X$ of schemes and an invertible $\mathcal{O}_ Y$-module $\mathcal{L}$ such that $F(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$.

Proof. Of course $F$ is additive, exact, commutes with all limits, commutes with all colimits, commutes with direct sums, etc. Let $U \subset X$ be an affine open subscheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a finite type quasi-coherent sheaf of ideals such that $Z = V(\mathcal{I})$ is the complement of $U$ in $X$, see Properties, Lemma 28.24.1. Then $\mathcal{O}_ X/\mathcal{I}$ is a finitely presented $\mathcal{O}_ X$-module. Hence $\mathcal{G} = F(\mathcal{O}_ X/\mathcal{I})$ is a finitely presented $\mathcal{O}_ Y$-module by Lemma 56.6.1. Denote $T \subset Y$ the support of $\mathcal{G}$ and set $V = Y \setminus T$. Since $\mathcal{G}$ is of finite presentation, the scheme $V$ is a quasi-compact open of $Y$. By Lemma 56.6.3 we see that $F$ induces an equivalence between

  1. the full subcategory of $\mathit{QCoh}(\mathcal{O}_ X)$ consisting of modules supported on $Z$, and

  2. the full subcategory of $\mathit{QCoh}(\mathcal{O}_ Y)$ consisting of modules supported on $T$.

By Lemma 56.6.4 we obtain a commutative diagram

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ X) \ar[r]_ F \ar[d] & \mathit{QCoh}(\mathcal{O}_ Y) \ar[d] \\ \mathit{QCoh}(\mathcal{O}_ U) \ar[r]^{F_ U} & \mathit{QCoh}(\mathcal{O}_ V) } \]

where the vertical arrows are the restruction functors and the horizontal arrows are equivalences. By Lemma 56.6.5 we conclude that $V$ is affine. For the affine case we have Lemma 56.3.8. Thus we find that there is an isomorphism $f_ U : V \to U$ and an invertible $\mathcal{O}_ V$-module $\mathcal{L}_ U$ such that $F_ U$ is the functor $\mathcal{F} \mapsto f_ U^*\mathcal{F} \otimes \mathcal{L}_ U$.

The proof can be finished by noticing that the diagrams above satisfy an obvious compatibility with regards to inclusions of affine open subschemes of $X$. Thus the morphisms $f_ U$ and the invertible modules $\mathcal{L}_ U$ glue. We omit the details. $\square$

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