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On the Ext-computability of Serre quotient categories (1212.4068v3)
Published 17 Dec 2012 in math.KT and math.AG
Abstract: 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 $\mathrm{Ext}$.