Consider a smooth hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $d$ over a nice field (such as $\mathbb{C}$), we know that the cone $C(\operatorname{id}:\mathcal{O}\_X\rightarrow\mathcal{O}\_X)=0$. Under the isomorphism
$$\hom(\mathcal{O}\_X,\mathcal{O}\_X)\cong\hom(\mathcal{O}\_X,\mathcal{O}\_X(d-n-2)[n])^\vee\cong\mathbb{C}$$
one can identify $\operatorname{id}$ with a complex map $$f：\mathcal{O}\_X\rightarrow\mathcal{O}\_X(d-n-2)[n].$$
Can we compute the cone $E$ of this map in $D^b(X)$? Explicitly, what are the cohomology sheaves of $E$?



Compute the cone of $\mathcal{O}_X\rightarrow\mathcal{O}_X(d-n-2)[n]$
The cohomology sheaves are easy to see from the long sequence of cohomology sheaves of the triangle
$$
\mathcal{O}\_X \to \mathcal{O}\_X(d-n-2)[n] \to E.
$$
If $n \ge 2$ it gives
$$
\mathcal{H}^i(E) = 
\begin{cases}
\mathcal{O}\_X(d-n-2), & i = -n,\\
\mathcal{O}\_X, & i = -1,\\
0, & \text{otherwise.}
\end{cases}
$$
And if $n = 1$ there is a single cohomology sheaf in the degree $-1$, and it is an extension of $\mathcal{O}\_X$ by $\mathcal{O}\_X(d-3)$.


