# Proof Explanation of Lemma III 7.4 Hartshorne

In Lemma III 7.4 of Hartshorne we have $$X$$ is a closed subscheme of co-dim $$r$$ in $$P=\mathbb P_k^N$$. $$\mathscr F$$ is a coherent sheaf on $$X$$. Then $$\mathscr F$$ is naturally an $$\mathcal O_P$$ module. We have to show

$$Hom_X (\mathscr F, \omega_X)=Ext ^r_P(\mathscr F,\omega_P)$$

Hartshorne takes an injective resolution of $$\omega_P$$ as an $$\mathcal O_P$$ module $$0\rightarrow\omega_P\rightarrow \mathcal {\mathcal I}^{\bullet}$$

Then Hartshorne says since $$\mathscr F$$ is an $$\mathcal O_X$$ module any morhpism $$\mathscr F \rightarrow \mathcal I^i$$ factors through $$\mathscr I^i :=\mathscr {Hom} _P(\mathcal O_X, \mathcal I^i)$$

I have a problem at this stage.

I know $$\mathscr F \simeq \mathscr F \otimes _{\mathcal O_P} \mathcal O_X$$ as $$\mathcal O_P$$ modules and hence we have via the adjunction formula $$Hom_P(\mathscr F, \mathcal I^i)\simeq Hom_P(\mathscr F, \mathscr {Hom}(O_X, \mathcal I^i))$$

Is it because of this reason. It is a little vague to me so I shall be highly obliged if someone spells out the details to me.

I think your reasoning is okay. Alternatively, let $$\mathcal{J}$$ denote the sheaf of ideals picking out $$X$$. Since the image of any morphism from a $$\mathcal{J}$$-torsion sheaf is again $$\mathcal{J}$$-torsion and $$\mathcal{O}_X$$ is $$\mathcal{J}$$-torsion, all we need to do is to identify the $$\mathcal{J}$$-torsion of $$\mathcal{I}^i$$ and then we know that our morphism factors through that (via the inclusion of the torsion in to $$\mathcal{I}^i$$). But this torsion is easy to figure out - the $$\mathcal{J}$$ torsion is just $$\mathcal{Hom}_P(\mathcal{O}_P/\mathcal{J},\mathcal{I}^i)=\mathcal{Hom}_P(\mathcal{O}_X,\mathcal{I}^i)$$, so the image of our map factors through this as Hartshorne says.