On the Ext-computability of Serre quotient categories

Understand

To develop a constructive description of $\mathrm{Ext}$ in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the $\mathrm{Ext}$-groups in Serre quotient categories $\mathcal{A}/\mathcal{C}$ and a direct limit of $\mathrm{Ext}$-groups in the ambient Abelian category $\mathcal{A}$.

• For $\mathrm{Ext}^1$ 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 $\mathcal{A}/\mathcal{C}$ to have enough projectives or injectives and therefore use Yoneda's description of $\mathrm{Ext}$.