assumption and notation as above, let $F$ be a coherent sheaf on $X$. Suppose that $\operatorname{Ext}^q(G,F)=0$ for all $q>0$. Then $F$ has a locally free resolution of the following form: $$0\to G_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}G_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}G_{j}^{\oplus e_{0,j}}\to F\to 0$$ 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: $$e_{l,j}=\sum_{j<k}e_{l-1,k}\dim \operatorname{Hom}(G_j,G_k).$$

Since $\operatorname{Ext}^q(G,F)=0$ for all $q>0$, Bondal’s theorem [@MR992977 Theorem 6.2] implies that $\operatorname{Hom}(G, F)\operatorname{\otimes^{\mathbb{L}}}_A G\cong F$. Since $\operatorname{Hom}(G,F)^j\cong \operatorname{Hom}(G_j,F)$, Lemma \[projresol\] shows the following projective resolution of $\operatorname{Hom}(G,F)$: $$0\to P_0^{\oplus e_{m,0}}\to\dots\to \bigoplus_{j=0}^{m-l}P_{j}^{\oplus e_{l,j}}\to\dots\to
\bigoplus_{j=0}^{m}P_{j}^{\oplus e_{0,j}}\to \operatorname{Hom}(G,F)\to 0.$$ Since $P_j\otimes_AG\cong G_j$, the projective resolution above induces the desired locally free resolution of the coherent sheaf $F$.

Set $\mathcal{P}_l=\oplus_{j=0}^{m-l}G_{j}^{\oplus e_{l,j}}$ for $0\leqq l\leqq m$, and let $\mathcal{P}_{\bullet}$ denote the resulting complex.

Set $\operatorname{HN}_i(\mathcal{P}_{\bullet})=
\oplus_{j=i}^{m-\bullet}
G_j^{\oplus e_{\bullet,j}}$. Then $\mathcal{P}_{\bullet}$ has the following filtration of Harder-Narasimhan type $$0\to \operatorname{HN}_m(\mathcal{P}_{\bullet})\to \dots \to \operatorname{HN}_i(\mathcal{P}_{\bullet})\to \dots\to \operatorname{HN}_0(\mathcal{P}_{\bullet})=\mathcal{P}_{\bullet}$$ with $\operatorname{HN}_{i}(\mathcal{P}_{\bullet})/\operatorname{HN}_{i+1}(\mathcal{P}_{\bullet})\cong 
\oplus_{i\leqq m-\bullet}G_i^{\oplus e_{\bullet,i}}\in \langle G_i\rangle
$, where $\langle G_i\rangle$ denotes the smallest triangulated subcategory of $D^b(X)$ containing $G_i$ and closed under isomorphism. Note that $\operatorname{Hom}_{D^b(X)}(\langle G_i\rangle,\langle G_j\rangle)=0$ if $i>j$. If we regard $\mathcal{P}_{\bullet}$ and $F$ as objects of $D^b(X)$ and the filtration above as 