On the Ext-computability of Serre quotient categories

To develop a constructive description of Ext in categories of coherent sheaves over certain schemes, we establish a binatural isomorphism between the Ext-groups in Serre quotient categories A/C and a direct limit of Ext-groups in the ambient Abelian category A. For Ext1 the isomorphism follows if the thick subcategory C⊂A is localizing. For the higher extension groups we need further assumptions on C. With these categories in mind we cannot assume A/C to have enough projectives or injectives and therefore use Yoneda's description of Ext.