I am studying section III.7 (The Serre Duality Theorem) of Hartshorne's Algebraic Geometry and have some issues with the proof of
Theorem 7.11. Let $X$ be a closed subscheme of $P = \mathbb{P}^N_k$ which is a local complete intersection of codimension $r$. Let $\mathscr{I}$ be the ideal sheaf of $X$. Then $\omega_X^{\circ} \cong \omega_P \otimes \bigwedge^r (\mathscr{I}/\mathscr{I}^2)^\vee$. In particular, $\omega_X^{\circ}$ is an invertible sheaf on $X$.
Say $j \colon X \hookrightarrow P$ is a closed immersion, defining the ideal sheaf \begin{align*} \mathscr{I} = \ker(j^{\flat} \colon \mathscr{O}_P \twoheadrightarrow j_* \mathscr{O}_X). \end{align*} I guess what we want to show is $j_* \omega^{\circ}_X \cong \omega_P \otimes \bigwedge^r (\mathscr{I}/\mathscr{I}^2)^\vee$. What I already know is that \begin{align*} \omega^{\circ}_X \cong j^* \mathscr{E}xt^r_P(j_* \mathscr{O}_X, \omega_P), \end{align*} so that \begin{align*} j_*\omega^{\circ}_X \cong j_*j^* \mathscr{E}xt^r_P(j_* \mathscr{O}_X, \omega_P) \cong \mathscr{E}xt^r_P(j_* \mathscr{O}_X , \omega_P). \end{align*} Hence we have to compute $\mathscr{E}xt_P^r(j_* \mathscr{O}_X , \omega_P)$. Now this is where Hartshorne's proof (and with it my problems) begins. Since I don't understand the way how Hartshorne gets the required isomorphism I just start writing down things I know (or think to know) - maybe, with someone's help, it turns out that this gives a rise to a way of proving the theorem in a way I understand. The aim is to cover $P$ by open affines $U_i \cong \text{Spec}(A_i)$, such that $\mathscr{I}_{|U_i}$ is generated by $r$ local sections $f_{i,1}, \dots , f_{i,r} \in \Gamma(U_i, \mathscr{O}_P)$ (this is possible since $X$ is a local complete intersection in $P$) and then to construct isomorphisms \begin{align*} \mathscr{E}xt_P(j_* \mathscr{O}_X, \omega_P)_{|U_i} \overset{\sim}{\longrightarrow} (\omega_P \otimes \bigwedge^r (\mathscr{I}/\mathscr{I}^2)^\vee)_{| U_i}, \end{align*} which after all can be glued together to give the required isomorphism. What I know about the left hand side: As $P$ is noetherian and both $j_* \mathscr{O}_X$ and $\omega_P$ are coherent we have \begin{align*} \mathscr{E}xt_P(j_* \mathscr{O}_X, \omega_P)_{|U_i} \cong \text{Ext}^r_{A_i}(M_i,N_i)^{\sim} \end{align*} for some finitely generated $A_i$-modules $M_i,N_i$, satisfying $(j_* \mathscr{O}_X)_{| U_i} \cong M_i^{\sim}$ and ${\omega_P}_{| U_i} \cong N_i^{\sim}$. (Can there something be said about the modules $M_i$ and $N_i$? As $j_* \mathscr{O}_X \cong \mathscr{O}_P / \mathscr{I}$ it feels like there could be something.) What I (think to) know about the right hand side: Using the fact that $\mathscr{I}/\mathscr{I}^2$ is locally free of finite rank we have \begin{align*} \omega_P \otimes \bigwedge^r (\mathscr{I}/\mathscr{I}^2)^{\vee} \cong \mathscr{H}om_{\mathscr{O}_P}(\bigwedge^r \mathscr{I}/\mathscr{I}^2 , \omega_P). \end{align*} Can we do something similar with this guy (i.e. writing the restriction of this as some tilde of a module)? Then it would look like we "only" had to give an isomorphism from some Ext to some Hom. (Well, and then think about glueing..) Thanks in advance for any comment.
What you're describing is a special case of the fundamental local isomorphism: for a lci morphism $i:Y\hookrightarrow X$ of pure codimension $n$ and any $\mathcal{O}_X$-module $\mathcal{F}$, we have that $\mathscr{Ext}^n_X(i_*\mathcal{O}_Y,\mathcal{F})\cong \omega_{Y/X} \otimes i^*(\mathcal{F})$ compatible with localization on $X$. The following treatment follows Brian Conrad's book on Grothendieck Duality and Base Change, available on his website.
To start, let us work affine-locally. Suppose $X=\operatorname{Spec} A$ with $Y\hookrightarrow X$ a closed subscheme given by the ideal $J$ which is generated by the regular sequence $(f_1,f_2,\cdots,f_n)$. Then $Y\hookrightarrow X$ is a lci morphism of pure codimension $n$.
For any $A$-module $M$, we may compute $$\operatorname{Ext}^i_A(A/J,M) = H^i(\operatorname{Hom}^\bullet_A(K_\bullet(\textbf{f}),M))$$ where $K_\bullet(\textbf{f})$ is the Koszul complex associated to the regular sequence $(f_1,\cdots,f_n)$. The key feature of $K_\bullet(\textbf{f})$ in this case is that it is a free resolution of $A/J$. This enables us to write $\operatorname{Ext}_A^i(A/J,M)=0$ except when $i=n$ and in that case we get $\operatorname{Ext}_A^n(A/J,M)=M/JM$. From here, we use the isomorphism $A/J \to \bigwedge^n(J/J^2)$ given by $1\mapsto f_1\wedge f_2\wedge\cdots\wedge f_n$ and the isomorphism $M/JM \otimes A/J \cong M/JM$ to get an isomorphism $M/JM \to \bigwedge^n(J/J^2)\otimes_{A/J} M/JM$. Composing these two isomorphisms, we see that we have an isomorphism $$\operatorname{Ext}_A^i(A/J,M)\to \bigwedge^n(J/J^2)^*\otimes_{A/J} M/JM$$ which is independent of the regular sequence we used, so we can sheafify and globalize to reach the result in the general case.