The Stacks project

Lemma 30.23.7. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Any object of $\textit{Coh}(X, \mathcal{I})$ which is annihilated by a power of $\mathcal{I}$ is in the essential image of ( Moreover, if $\mathcal{F}$, $\mathcal{G}$ are in $\textit{Coh}(\mathcal{O}_ X)$ and either $\mathcal{F}$ or $\mathcal{G}$ is annihilated by a power of $\mathcal{I}$, then the maps

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{F}, \mathcal{G}) \ar[d] & \mathop{\mathrm{Ext}}\nolimits _ X(\mathcal{F}, \mathcal{G}) \ar[d] \\ \mathop{\mathrm{Hom}}\nolimits _{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge , \mathcal{G}^\wedge ) & \mathop{\mathrm{Ext}}\nolimits _{\textit{Coh}(X, \mathcal{I})}(\mathcal{F}^\wedge , \mathcal{G}^\wedge ) } \]

are isomorphisms.

Proof. Suppose $(\mathcal{F}_ n)$ is an object of $\textit{Coh}(X, \mathcal{I})$ which is annihilated by $\mathcal{I}^ c$ for some $c \geq 1$. Then $\mathcal{F}_ n \to \mathcal{F}_ c$ is an isomorphism for $n \geq c$. Hence if we set $\mathcal{F} = \mathcal{F}_ c$, then we see that $\mathcal{F}^\wedge \cong (\mathcal{F}_ n)$. This proves the first assertion.

Let $\mathcal{F}$, $\mathcal{G}$ be objects of $\textit{Coh}(\mathcal{O}_ X)$ such that either $\mathcal{F}$ or $\mathcal{G}$ is annihilated by $\mathcal{I}^ c$ for some $c \geq 1$. Then $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$ is a coherent $\mathcal{O}_ X$-module annihilated by $\mathcal{I}^ c$. Hence we see that

\[ \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = H^0(X, \mathcal{H}) = \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ). \]

see Lemma 30.23.5. This proves the statement on homomorphisms.

The notation $\mathop{\mathrm{Ext}}\nolimits $ refers to extensions as defined in Homology, Section 12.6. The injectivity of the map on $\mathop{\mathrm{Ext}}\nolimits $'s follows immediately from the bijectivity of the map on $\mathop{\mathrm{Hom}}\nolimits $'s. For surjectivity, assume $\mathcal{F}$ is annihilated by a power of $I$. Then part (1) of Lemma 30.23.6 shows that given an extension

\[ 0 \to \mathcal{G}^\wedge \to (\mathcal{E}_ n) \to \mathcal{F}^\wedge \to 0 \]

in $\textit{Coh}(U, I\mathcal{O}_ U)$ the morphism $\mathcal{G}^\wedge \to (\mathcal{E}_ n)$ is isomorphic to $\mathcal{G} \to \mathcal{E}^\wedge $ for some $\mathcal{G} \to \mathcal{E}$ in $\textit{Coh}(\mathcal{O}_ U)$. Similarly in the other case. $\square$

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