Proposition 30.25.4. Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Set $\mathcal{I} = I\mathcal{O}_ X$. Then the functor (30.23.3.1) is an equivalence.

Proof. We have already seen that (30.23.3.1) is fully faithful in Lemma 30.24.1. Thus it suffices to show that the functor is essentially surjective.

Consider the collection $\Xi$ of quasi-coherent sheaves of ideals $\mathcal{K} \subset \mathcal{O}_ X$ such that every object $(\mathcal{F}_ n)$ annihilated by $\mathcal{K}$ is in the essential image. We want to show $(0)$ is in $\Xi$. If not, then since $X$ is Noetherian there exists a maximal quasi-coherent sheaf of ideals $\mathcal{K}$ not in $\Xi$, see Lemma 30.10.1. After replacing $X$ by the closed subscheme of $X$ corresponding to $\mathcal{K}$ we may assume that every nonzero $\mathcal{K}$ is in $\Xi$. (This uses the correspondence by coherent modules annihilated by $\mathcal{K}$ and coherent modules on the closed subscheme corresponding to $\mathcal{K}$, see Lemma 30.9.8.) Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. We will show that this object is in the essential image of the functor (30.23.3.1), thereby completion the proof of the proposition.

Apply Chow's lemma (Lemma 30.18.1) to find a proper surjective morphism $f : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X'$ is projective over $A$. Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By the projective case of Grothendieck's existence theorem (Lemma 30.24.3) there exists a coherent module $\mathcal{F}'$ on $X'$ such that $(\mathcal{F}')^\wedge \cong (f^*\mathcal{F}_ n)$. By Proposition 30.19.1 the $\mathcal{O}_ X$-module $\mathcal{H} = f_*\mathcal{F}'$ is coherent and by Lemma 30.25.3 there exists a morphism $(\mathcal{F}_ n) \to \mathcal{H}^\wedge$ of $\textit{Coh}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. The powers $\mathcal{K}^ e$ are all in $\Xi$ so that (30.23.3.1) is an equivalence for the closed subschemes $X_ e = V(\mathcal{K}^ e)$. We conclude by Lemma 30.25.2. $\square$

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