Lemma 30.23.5. Let $X$ be a Noetherian scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$. Then

**Proof.**
To prove this we may work affine locally on $X$. Hence we may assume $X = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{F}$, $\mathcal{G}$ given by finite $A$-module $M$ and $N$. Then $\mathcal{H}$ corresponds to the finite $A$-module $H = \mathop{\mathrm{Hom}}\nolimits _ A(M, N)$. The statement of the lemma becomes the statement

via the equivalence of Lemma 30.23.1. By Algebra, Lemma 10.97.2 (used 3 times) we have

where the second equality uses that $A^\wedge $ is flat over $A$ (see More on Algebra, Lemma 15.65.4). The lemma follows. $\square$

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