The Stacks project

Lemma 30.24.3. Let $A$ be Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a projective morphism. Let $\mathcal{I} = I\mathcal{O}_ X$. Then the functor ( is an equivalence.

Proof. We have already seen that ( is fully faithful in Lemma 30.24.1. Thus it suffices to show that the functor is essentially surjective.

We first show that every object $(\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ is the quotient of an object in the image of ( Let $\mathcal{L}$ be an $f$-ample invertible sheaf on $X$. Choose $d_0$ as in Lemma 30.24.2. Choose a $d \geq d_0$ such that $\mathcal{F}_1 \otimes \mathcal{L}^{\otimes d}$ is globally generated by some sections $s_{1, 1}, \ldots , s_{t, 1}$. Since the transition maps of the system

\[ H^0(X, \mathcal{F}_{n + 1} \otimes \mathcal{L}^{\otimes d}) \longrightarrow H^0(X, \mathcal{F}_ n \otimes \mathcal{L}^{\otimes d}) \]

are surjective by the vanishing of $H^1$ we can lift $s_{1, 1}, \ldots , s_{t, 1}$ to a compatible system of global sections $s_{1, n}, \ldots , s_{t, n}$ of $\mathcal{F}_ n \otimes \mathcal{L}^{\otimes d}$. These determine a compatible system of maps

\[ (s_{1, n}, \ldots , s_{t, n}) : (\mathcal{L}^{\otimes -d})^{\oplus t} \longrightarrow \mathcal{F}_ n \]

Using Lemma 30.23.3 we deduce that we have a surjective map

\[ \left((\mathcal{L}^{\otimes -d})^{\oplus t}\right)^\wedge \longrightarrow (\mathcal{F}_ n) \]

as desired.

The result of the previous paragraph and the fact that $\textit{Coh}(X, \mathcal{I})$ is abelian (Lemma 30.23.2) implies that every object of $\textit{Coh}(X, \mathcal{I})$ is a cokernel of a map between objects coming from $\textit{Coh}(\mathcal{O}_ X)$. As ( is fully faithful and exact by Lemmas 30.24.1 and 30.23.4 we conclude. $\square$

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