extremal ray of the Fano manifold $\mathbb{P}(\mathcal{E})$. On the other hand, our proof depends only on the cohomological study of $\mathcal{E}$ with respect to a full strong exceptional sequence of vector bundles, and does not analyze any contraction morphism. (See Theorems \[d=1onProSpace\], \[d2OnProSpace\], and \[d=1OnHyperquadric\] and the proofs therein.)

More precisely our framework is based on the following observation. Suppose there exists a full strong exceptional sequence $G_0,\dots,G_m$ of vector bundles on $X$. Recall here that a projective space or a smooth hyperquadric admits such a full strong exceptional sequences of vector bundles by [@MR0509388] and [@MR0939472]. Denote by $G$ the direct sum $\bigoplus _{i=0}^m G_i$, and by $A$ the endomorphism ring $\operatorname{End}(G)$ of $G$. Let $F$ be a coherent sheaf on $X$. Then Bondal’s theorem [@MR992977 Theorem 6.2] implies an isomorhism $$\operatorname{RHom}(G,F)\operatorname{\otimes^{\mathbb{L}}}_A G\cong F.$$ Suppose first that $\operatorname{Ext}^q(G,F)=0$ for all $q>0$. Then $\operatorname{RHom}(G,F)\cong \operatorname{Hom}(G,F)$, and the isomorphism above implies that a projective resolution of the right $A$-module $\operatorname{Hom}(G,F)$ induces the following locally free resolution of $F$: $$0\to G_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}G_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}G_{j}^{\oplus e_{0,j}}\to F\to 0$$ where $e_{0,j}=\dim \operatorname{Hom}(G_j,F)$ for all $j=0,\dots,m$ and, for any $l\geqq 1$ and any $j\leqq m-l$, $e_{l,j}$ is determined inductively by the following formula: $e_{l,j}=\sum_{j<k}e_{l-1,k}\dim 