that in $D^b(X)$, then $\mathcal{P}_{\bullet}\cong F$ and semiorthogonality of $\langle G_0\rangle,\dots,\langle G_m\rangle$ implies that the filtration above is unique and functorial with respect to $F$.

Let $\mathcal{E}$ be a coherent sheaf on $X$, and let $\mathcal{O}_X(1)$ be an ample line bundle on $X$. By Serre’s vanishing theorem, we see that if $d'\gg 0$ then $\operatorname{Ext}^q(G,\mathcal{E}(d'))=0$ for all $q>0$. 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 $d'\geqq d_{\min}$.

\[Resol\] Under the notation above, $\mathcal{E}(d_{\min})$ fits in the following exact sequence: $$0\to G_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}G_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}G_{j}^{\oplus e_{0,j}}\to \mathcal{E}(d_{\min})\to 0,$$ where $e_{0,j}=\dim \operatorname{Hom}(G_j,\mathcal{E}(d_{\min}))$ 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 $e_{l,j}=\sum_{j<k}e_{l-1,k}\dim \operatorname{Hom}(G_j,G_k)$. Moreover we can replace $e_{l,j}$ by some integer $e_{l,j}'$ such that $0\leqq e_{l,j}'\leqq e_{l,j}$ for all $0\leqq j\leqq m-l\leqq m$ corresponding to the minimal resolution (see Remark \[MinimalResol\] for the precise definition of $e_{l,j}'$).

In the rest of this paper, we call the exact sequence in Corollary \[Resol\] *the standard resolution of $\mathcal{E}(d_{\min})$ with respect to the (prescribed) full strong exceptional sequence $(G_0,\dots,G_m)$ of vector bundles*, and let $e_{l,j}'$ and $e_{l,j}$ be as in Corollary \[Resol\]. The following is an easy but 