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.

  • $\begingroup$ Thanks for your answer! I will take a look at that. $\endgroup$ – Jupp Sep 12 '18 at 8:00
  • $\begingroup$ Okay so now I have the isomorphism in your last "align" (I guess one wants an extra dual on the wedge product, right?) and I know that is is independent of the choice of the regular sequence generating J. I also know that this gives me an isomorphism between the restrictions I mentioned. What is not quite clear to me is the glueing of these - can you tell me why they glue to give an isomorphism between the sheaves we are interested in? $\endgroup$ – Jupp Sep 14 '18 at 9:52
  • $\begingroup$ Yes, good catch - I mistyped and forgot the dual on the wedge product. The reason this isomorphism gives an isomorphism between the sheaves is that this isomorphism sheafifies to give an isomorphism between these objects on the affine scheme $\operatorname{Spec} A$, and that the constructed isomorphism is compatible with localization. This implies that the isomorphism of sheaves restricted to any open subset of $\operatorname{Spec} A$ matches the isomorphism of the sheaves defined on that open subset. Thus we get compatibility across different affine charts and we can glue. $\endgroup$ – KReiser Sep 14 '18 at 18:03
  • $\begingroup$ What is explicitly meant by "compatible with localization"? Maybe it is because I am not very familiar with this, but I don't understand "... matches the isomorphism defined on that open subset" - I thought we only defined isomorphisms on these open affines.. Maybe you could elaborate some more details on this glueing process.. Thanks a lot in advance for your (already given) help! $\endgroup$ – Jupp Sep 21 '18 at 10:31
  • $\begingroup$ Compatible with localization: for every open affine set, we construct a pair of sheaves and an isomorphism between them. It turns out that if $U\subset V$ are both affine open, the sheaves and morphism we get from considering those over $V$ restricted to $U$ match what we would build over $U$. This tells us that the cocycle condition is satisfied, so that we may glue everything together. $\endgroup$ – KReiser Sep 23 '18 at 1:36

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