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

$\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

$a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

$\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and

$\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

$\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

$a : \mathcal{G} \to \mathcal{F}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

$\beta : \mathcal{F}^\wedge \to (\mathcal{F}_ n)$ is an isomorphism, and

$\alpha = \beta \circ a^\wedge $.

## Comments (0)