Construction of Dualizing sheaf

Question

I was going through the construction of dualizing sheaf given in Hartshorne [III, 7, Lemma 7.4]. The proof apparently omits lots of details. In particular it does not mention any of the $i^* , i_*, i^{!}$ that should appear, so almost none of the things he writes are actually defined. I was trying to clean up the proof and I noticed the following more general algebraic lemma.

Lemma: Let $\mathcal C, \mathcal D$ be abelian categories and $\mathcal D$ has enough injectives. $F : \mathcal C \to \mathcal D$ and $G:\mathcal D \to \mathcal C $ are additive functors such that

(i) $F$ is left exact and sends nonzero objects to nonzero objects.

(ii) $G$ is right adjoint to $F$.

(ii) For some fixed object $P$ of $\mathcal D$, we have $(R^i (F \circ G) )(P) = 0$ for all $i < r$ for some fixed $r \in \mathbb N$.

Then it follows that for any object $A$ of $\mathcal C$, $\text{Ext}^i (F(A), P) = 0$ for $i < r$ and there is a functorial isomorphism $\text{Ext}^r (F(A), P) \cong \text{Hom}(A, (R^r (G))(P))$.

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Proof: We choose an injective resolution of $P$ in $\mathcal D$, $0 \to P \to I^{\bullet}$. Define $J^{\bullet} := G(I^{\bullet})$. Since $F$ is left exact and left adjoint it is in fact exact. By adjointness, $J^{\bullet}$ is a complex of injective objects. Now $F(h^iJ^{\bullet}) = h^i( F(J^{\bullet}))= h^i (F \circ G )(I^{\bullet}) = R^i(F \circ G)= 0$ for $i<r$ by hypothesis. So $J^{\bullet}$ is exact till $r$-th step. Therefore we can write $J^{\bullet}$ as $J_1^{\bullet} \oplus J_2^{\bullet}$, where $J_1^{\bullet}$ is injective, exact and nonzero only in degrees $\le r$ and $J_2$ is also injective and nonzero only in degrees $\ge r$. Now $\text{Ext}^r(F(A), P) = h^r (\text{Hom}(F(A), I^{\bullet})= h^r(\text{Hom}(A, J^{\bullet}))$ $=h^r(\text{Hom}(A, J_2^{\bullet})) = $ $\text{Ker}((\text{Hom}(A, J_2^r) \to \text{Hom}(A, J_2^{r+1}))$ $ = \text{Hom}(A, \text{Ker}(J_2^{r} \to J_2^{r+1}))$. But $\text{Ker}(J_2^{r} \to J_2^{r+1}) = h^r(J^{\bullet})= h^r(G(I^{\bullet}))= (R^rG)(P)$.

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Now to prove the existence of dualizing sheaf for a closed immersion $j : X \to \mathbb{P}^n_k$, we take $\mathcal C = \mathfrak {Qco} (X), \mathcal D = \mathfrak {Qco} (\mathbb P ^n _k), F = j_*, G = j^{!}$. Then $(R^i (j_* \circ j^{!})) (\omega_{\mathbb{P}^n_k}) = \mathscr{Ext}^i(j_* \mathscr{O}_X, \omega_{\mathbb{P}^n_k}) = 0$ for $i < r:= n - \dim X$. Therefore, it follows from lemma that $\text{Ext}^r(j_*\mathscr{F}, \omega_{\mathbb{P}^n_k}) = \text{Hom}_{X}(\mathscr{F}, (R^{r} (j^!))(\omega_{\mathbb{P}^n_k}) )$. Now $(R^{r} (j^!))(\omega_{\mathbb{P}^n_k})$ is same as the quasicoherent $\mathcal{O}_X$- module corresponding to the quasicoherent $j_* \mathcal{O}_X$ module $\mathscr{Ext}^r (j_* \mathcal O _{X}, \omega_{\mathbb{P}^n_k})$, which we write by $\mathscr{Ext}^r (j_* \mathcal O _{X}, \omega_{\mathbb{P}^n_k})^{\sim}$.

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Questions:

(i)Does the lemma follow quickly from some spectral sequence (which I don't properly know) argument? Is there any reference for the lemma?

(ii)In these notes (http://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/lecture-notes/MIT18_726s09_lec24_dualizing.pdf), the dualizing sheaf is defined to be $j^*(\mathscr{Ext}^r (j_* \mathcal{O}_X, \omega_{\mathbb{P}^n_k}))$. Is it the same as $\mathscr{Ext}^r (j_* \mathcal{O}_X, \omega_{\mathbb{P}^n_k})^{\sim}$?

I don't really understand some parts of the proof in that note, in particular it talks about isomorphism of sheaves in two different topological spaces, which should really be related by a push forward.

Answer 1

I am not sure I see the point of your worry. As $i_*$ is exact, there is really no harm in ignoring it. Hartshorne actually says this earlier and states that this "abuse of notation" will be used. As far as I can tell, $i^*$ is only used (that is "should be used, but not marked") when a sheaf supported on $X$, but considered as one on $\mathbb P^n$ is restricted to $X$. This is just using the previous principle. In other words, Hartshorne (and pretty much everyone else) adopts the maxim that

If the sheaf $\mathscr F$ on a scheme $Z$ is supported on the closed subscheme $X\subseteq Z$, then it may be considered a sheaf on $X$.

This is about the same as when a module $M$ over the ring $A$ has annihilator $I\subseteq A$, then $M$ maybe considered a module over $A/I$.

As for your lemma, it seems to me that it is indeed relatively straightforward from a Grothendieck spectral sequence argument: As you point out, (i) and (ii) imply that $F$ is actually exact (and of course, $i_*$ is, so you could even start with this). Then (i) and (iii) imply that you actually have
(iii*) $R^i G (P) = 0$ for all $i < r$ for some fixed $r \in \mathbb N$. Then considering the fact that $$R\text{Hom}(F(A), \_\_)\cong R\text{Hom}(A, G(\_\_)),$$ and applying GSS to the RHS, and pluggin in $P$ you see that by (iii*) $$E^{p,q}_2=R^p\text{Hom}(A, R^qG(P))=0$$ as long as $q<r$ (and of course it is $0$ for $p<0$), so the first non-zero term from the SW ("the lower left corner") of the spectral sequence computing the RHS is $\text{Hom}(A, R^rG(P))$. This proves both statements.