\operatorname{Hom}(G_j,G_k)$. For an arbitrary coherent sheaf $F$, we have $\operatorname{Ext}^q(G,F(d))=0$ for all $q>0$ if $d$ is sufficiently large by Serre’s vanishing. Therefore we have a resolution of $F$ of the above form by replacing $(G_0,\dots, G_m)$ by a new full strong exceptional sequence $(G_0(-d),\dots,G_m(-d))$. Note here that Serre’s vanishing does not give, in general, an effective estimate of the integer $d$. However, on a projective space or a smooth hyperquadric, the full strong exceptional sequence $G_0,\dots,G_m$ can be chosen to be that of well-understood vector bundles so that, by applying the Kodaira or Kawamata–Viehweg vanishing theorem, we can give an *effective* estimate of the integers $d$ such that $\operatorname{Ext}^q(G,\mathcal{E}(d))=0$ for all $q>0$ for a *nef* vector bundle $\mathcal{E}$ (Corollaries \[ProSpaceResol\], \[quadricResol3\], and \[quadricResol\]).

Let us look at our proof more closely, e.g., in the case where $X$ is an odd dimensional smooth complex hyperquadric $\mathbb{Q}^n$; in this case, we can take the sequence $(G_0,\dots,G_m)$ to be $(\mathcal{O},\mathcal{S},\mathcal{O}(1), \dots,\mathcal{O}(n-1))$ where $\mathcal{S}$ is the (spanned) spinor bundle. For a nef vector bundle $\mathcal{E}$ on $X$, let $d_{\min}$ be the minimal integer $d_{\min}$ such that $\operatorname{Ext}^q(G,\mathcal{E}(d'))=0$ for all $q>0$ and all $d'\geqq d_{\min}$. Then the Kodaira vanishing theorem shows that $d_{\min}\leqq c_1(\mathcal{E})$ (Corollary 