Let $Y$ be a smooth cubic 4-fold in $\mathbf{P}^5$. The derived category of $Y$ admits a semiorthogonal decomposition 


$$D^b(Y) = \langle \mathcal{A}\_Y, \mathcal{O}\_Y, \mathcal{O}\_{Y}(1), \mathcal{O}\_{Y}(2) \rangle,$$


where $\mathcal{A}\_Y$ is the right-orthogonal to the triangulated subcategory generated by $\mathcal{O}\_Y, \mathcal{O}\_{Y}(1), \mathcal{O}\_{Y}(2)$. Namely, $\mathcal{A}\_Y$ is the full subcategory with objects those $F \in D^b(Y)$ such that $Hom\_{D^b(Y)}(\mathcal{O}\_Y(j), F[i]) = 0$ for $j=0,1,2$ and all $i$.


Kuznetsov studies $\mathcal{A}\_Y$ in his paper "Derived categories of cubic fourfolds" <http://arxiv.org/abs/0808.3351>. He parenthetically remarks that the Serre functor of $\mathcal A\_Y$ is the shift by two functor $[2]$.


The question is: How do you show this?


The only reference I could find for this result is Corollary 4.4 of the paper <http://arxiv.org/abs/math/0303037>. However, I don't understand the proof given there. Specifically, I don't know how to justify the following points in Lemma 4.2 preceding the Corollary (notation as in the paper):


1) How do you get the expression Kuznetsov gives for the kernel of $O^d$ from the kernel of $O$?


2) Why is the restriction to $Y \times Y$ of the Beilinson resolution of the diagonal in $\mathbf{P}^{n+1} \times \mathbf{P}^{n+1}$ quasi-isomorphic to ${\Delta\_{Y}}\_\*(\mathcal O\_{Y})$? It seems the paper is being sloppy here with what degrees the complexes live in: Let's call $A$ the restriction of the Beilinson resolution (including the final term ${\Delta\_{Y}}\_\*(\mathcal O\_{Y}(d))$ as in the paper). Let's place A in nonpositive degrees so that ${\Delta\_{Y}}\_\*(\mathcal O\_{Y}(d))$ is in degree $0$ (I assume this is what is intended in the paper). Then I think the precise statement should be that $A$ is quasi-isomorphic to ${\Delta\_{Y}}\_\*(\mathcal O\_{Y})[2]$. Moreover, since $A$ is the restriction of a resolution of $\Delta\_\*(\mathcal O\_{\mathbf{P}^{n+1}}(d))$ by locally free sheaves, this can be rephrased as the following computation of left-derived functors: if $i : Y \times Y \hookrightarrow \mathbf{P}^{n+1} \times \mathbf{P}^{n+1}$ is the inclusion, then
$$L\_1 i^\*(\Delta\_\*(\mathcal O\_{\mathbf{P}^{n+1}}(d))) = {\Delta\_{Y}}\_\*(\mathcal O\_{Y}) $$
and
$$L\_k i^\*(\Delta\_\*(\mathcal O\_{\mathbf{P}^{n+1}}(d))) = 0$$
for $k > 1$.
How do you show this?



Serre functor of orthogonal subcategory of derived category of cubic 4-fold
The first is a straightforward computation. For example, to compute $O^2$ you note that the convolution of kernels preserves exactness of the triangles, hence one has distinguished triangles
$$
\Delta\_{Y\*}O\_Y(1) \circ K\_1 \to O\_Y(2)\boxtimes O\_Y \to \Delta\_\*O\_Y(2),
$$
$$
(O\_Y(1)\boxtimes O\_Y)\circ K\_1 \to V^\*\otimes O\_Y(1)\boxtimes O \to O\_Y(1)\boxtimes O\_Y(1),
$$
(where $V^\* = H^0(O\_Y(1))$), and
$$
K\_1\circ K\_1 \to (O\_Y(1)\boxtimes O\_Y)\circ K\_1 \to \Delta\_{Y\*}O\_Y(1) \circ K\_1.
$$
The second triangle implies that $(O\_Y(1)\boxtimes O\_Y)\circ K\_1 \cong O\_Y(1)\boxtimes \Omega(1)\_{|Y}$, so substituting this and the first triangle into the third triangle you get an expression for $O^2$. Then you repeat this computation.


For the second you need to compute $L(\alpha\times\alpha)^\*\Delta\_\*O$ (in the notation of the paper). Let us first compute $(\alpha\times\alpha)\_\*L(\alpha\times\alpha)^\*\Delta\_\*O$ instead. By projection formula it is isomorphic to
$$
(\alpha\times\alpha)\_\*L(\alpha\times\alpha)^\*\Delta\_\*O \cong (\alpha\times\alpha)\_\*O\_{Y\times Y} \otimes^L \Delta\_\*O\_P.
$$
To compute this we use the resolution
$$
0 \to O(-d)\boxtimes O(-d) \to O(-d)\boxtimes O \oplus O \boxtimes O(-d) \to O \boxtimes O
$$
of $(\alpha\times\alpha)\_\*O\_{Y\times Y}$. Tensoring it with $\Delta\_\*O\_P$ one gets the complex
$$
0 \to \Delta\_\*O\_P(-2d) \to \Delta\_\*O\_P(-d) \oplus \Delta\_\*O\_P(-d) \to \Delta\_\*O\_P
$$
which is a direct sum od the resolution of $\Delta\_\*O\_Y$ and of $\Delta\_\*O\_Y(-d)[1]$. Thus
$$
(\alpha\times\alpha)\_\*L(\alpha\times\alpha)^\*\Delta\_\*O \cong \Delta\_\*O\_Y \oplus \Delta\_\*O\_Y(-d)[1].
$$
Finally, since $\alpha\times\alpha$ is a closed embedding, the functor $(\alpha\times\alpha)\_\*$ is exact and conservative, hence the claim.


