strong exceptional sequence of vector bundles, where $\mathcal{S}^+$ and $\mathcal{S}^-$ are spinor bundles, and it is full by Kapranov’s theorem [@MR0939472 Theorem 4.10]. Recall here that we follow Kapranov’s convention for “spinor bundles”; for example, on a smooth quadric surface $\mathbb{Q}^2$, spanned line bundles $\mathcal{O}(1,0)$ and $\mathcal{O}(0,1)$ are spinor bundles, and thus $(\mathcal{O},\mathcal{O}(1,0),\mathcal{O}(0,1),\mathcal{O}(1))$ is a full strong exceptional sequence on $\mathbb{Q}^2$.

For some other fundamental facts about derived categories, we refer to an excellent book [@MR2182076] of Kashiwara-Schapira as a literature written in English.

Denote by $G$ the direct sum $\bigoplus_{i=0}^mG_i$ of $G_0,\dots,G_m$, and by $A$ the endomorphism ring $\operatorname{End}(G)$ of $G$. Then $A$ is a finite-dimensional $K$-algebra. We refer to [@MR2197389 Chap. I, II, III] for some basic facts about modules over a finite-dimensional $K$-algebra. Note that we follow the convention that the composite of two arrows $\alpha:a\to b$ and $\beta:b\to c$ is denoted by $\beta\alpha$. In the same vein, we regard $G$ as a left $A$-module.

For a coherent sheaf $F$ on $X$, Bondal’s theorem [@MR992977 Theorem 6.2] implies that $$\operatorname{RHom}(G, F)\operatorname{\otimes^{\mathbb{L}}}_A F\cong F,$$ so that if $\operatorname{Ext}^q(G,F)=0$ for all $q>0$ then $\operatorname{RHom}(G, F)\cong \operatorname{Hom}(G, F)$, and a projective resolution of the right $A$-module $\operatorname{Hom}(G, F)$ will play 