Lemma 30.25.2. Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. Let $X_ e \subset X$ be the closed subscheme cut out by $\mathcal{K}^ e$. Let $\mathcal{I}_ e = \mathcal{I}\mathcal{O}_{X_ e}$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. Assume
the functor $\textit{Coh}(\mathcal{O}_{X_ e}) \to \textit{Coh}(X_ e, \mathcal{I}_ e)$ is an equivalence for all $e \geq 1$, and
there exists a coherent sheaf $\mathcal{H}$ on $X$ and a map $\alpha : (\mathcal{F}_ n) \to \mathcal{H}^\wedge $ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.
Then $(\mathcal{F}_ n)$ is in the essential image of (30.23.3.1).
Proof.
During this proof we will use without further mention that for a closed immersion $i : Z \to X$ the functor $i_*$ gives an equivalence between the category of coherent modules on $Z$ and coherent modules on $X$ annihilated by the ideal sheaf of $Z$, see Lemma 30.9.8. In particular we may identify $\textit{Coh}(\mathcal{O}_{X_ e})$ with the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{K}^ e$ and $\textit{Coh}(X_ e, \mathcal{I}_ e)$ as the full subcategory of $\textit{Coh}(X, \mathcal{I})$ of objects annihilated by $\mathcal{K}^ e$. Moreover (1) tells us these two categories are equivalent under the completion functor (30.23.3.1).
Applying this equivalence we get a coherent $\mathcal{O}_ X$-module $\mathcal{G}_ e$ annihilated by $\mathcal{K}^ e$ corresponding to the system $(\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$. The maps $\mathcal{F}_ n/\mathcal{K}^{e + 1}\mathcal{F}_ n \to \mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n$ correspond to canonical maps $\mathcal{G}_{e + 1} \to \mathcal{G}_ e$ which induce isomorphisms $\mathcal{G}_{e + 1}/\mathcal{K}^ e\mathcal{G}_{e + 1} \to \mathcal{G}_ e$. Hence $(\mathcal{G}_ e)$ is an object of $\textit{Coh}(X, \mathcal{K})$. The map $\alpha $ induces a system of maps
\[ \mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n \longrightarrow \mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H} \]
whence maps $\mathcal{G}_ e \to \mathcal{H}/\mathcal{K}^ e\mathcal{H}$ (by the equivalence of categories again). Let $t \geq 1$ be an integer, which exists by assumption (2), such that $\mathcal{K}^ t$ annihilates the kernel and cokernel of all the maps $\mathcal{F}_ n \to \mathcal{H}/\mathcal{I}^ n\mathcal{H}$. Then $\mathcal{K}^{2t}$ annihilates the kernel and cokernel of the maps $\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n \to \mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H}$, see Remark 30.25.1. Whereupon we conclude that $\mathcal{K}^{4t}$ annihilates the kernel and the cokernel of the maps
\[ \mathcal{G}_ e \longrightarrow \mathcal{H}/\mathcal{K}^ e\mathcal{H}, \]
see Remark 30.25.1. We apply Lemma 30.23.6 to obtain a coherent $\mathcal{O}_ X$-module $\mathcal{F}$, a map $a : \mathcal{F} \to \mathcal{H}$ and an isomorphism $\beta : (\mathcal{G}_ e) \to (\mathcal{F}/\mathcal{K}^ e\mathcal{F})$ in $\textit{Coh}(X, \mathcal{K})$. Working backwards, for a given $n$ the triple $(\mathcal{F}/\mathcal{I}^ n\mathcal{F}, a \bmod \mathcal{I}^ n, \beta \bmod \mathcal{I}^ n)$ is a triple as in the lemma for the morphism $\alpha _ n \bmod \mathcal{K}^ e : (\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n) \to (\mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H})$ of $\textit{Coh}(X, \mathcal{K})$. Thus the uniqueness in Lemma 30.23.6 gives a canonical isomorphism $\mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n$ compatible with all the morphisms in sight. This finishes the proof of the lemma.
$\square$
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