Lemma 30.23.6 (0889)—The Stacks project

Lemma 30.23.6. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module. Let $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$.

If $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge $ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where
1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,
2. $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,
3. $\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and
4. $\alpha = a^\wedge \circ \beta $.
If $\alpha : \mathcal{G}^\wedge \to (\mathcal{F}_ n)$ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$, then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where
1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,
2. $a : \mathcal{G} \to \mathcal{F}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,
3. $\beta : \mathcal{F}^\wedge \to (\mathcal{F}_ n)$ is an isomorphism, and
4. $\alpha = \beta \circ a^\wedge $.

Proof. Proof of (1). The uniqueness implies it suffices to construct $(\mathcal{F}, a, \beta )$ Zariski locally on $X$. Thus we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{I}$ corresponds to the ideal $I \subset A$. In this situation Lemma 30.23.1 applies. Let $M'$ be the finite $A^\wedge $-module corresponding to $(\mathcal{F}_ n)$. Let $N$ be the finite $A$-module corresponding to $\mathcal{G}$. Then $\alpha $ corresponds to a map

\[ \varphi : M' \longrightarrow N^\wedge \]

whose kernel and cokernel are annihilated by $I^ t$ for some $t$. Recall that $N^\wedge = N \otimes _ A A^\wedge $ (Algebra, Lemma 10.97.1). By More on Algebra, Lemma 15.89.17 there is an $A$-module map $\psi : M \to N$ whose kernel and cokernel are $I$-power torsion and an isomorphism $M \otimes _ A A^\wedge = M'$ compatible with $\varphi $. As $N$ and $M'$ are finite modules, we conclude that $M$ is a finite $A$-module, see More on Algebra, Remark 15.89.20. Hence $M \otimes _ A A^\wedge = M^\wedge $. We omit the verification that the triple $(M, N \to M, M^\wedge \to M')$ so obtained is unique up to unique isomorphism.

The proof of (2) is exactly the same and we omit it. $\square$

The Stacks project

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