On the Ext-computability of Serre quotient categories

To develop a constructive description of $\mathrm{E}\mathrm{x}\mathrm{t}$ in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the $\mathrm{E}\mathrm{x}\mathrm{t}$-groups in Serre quotient categories $\mathcal{A}\mathrm{/}\mathcal{C}$ and a direct limit of $\mathrm{E}\mathrm{x}\mathrm{t}$-groups in the ambient Abelian category $\mathcal{A}$. For ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ the isomorphism follows if the thick subcategory $\mathcal{C}\subset \mathcal{A}$ is localizing. For the higher extension groups we need further assumptions on $\mathcal{C}$. With these categories in mind we cannot assume $\mathcal{A}\mathrm{/}\mathcal{C}$ to have enough projectives or injectives and therefore use Yoneda’s description of $\mathrm{E}\mathrm{x}\mathrm{t}$.