On the Ext-computability of Serre quotient categories

TL;DR

This paper establishes a constructive method to compute Ext groups in Serre quotient categories by relating them to direct limits of Ext groups in the ambient Abelian category, under certain conditions.

Contribution

It provides a binatural isomorphism between Ext groups in Serre quotients and direct limits in the ambient category, extending computability in categories lacking enough projectives or injectives.

Findings

Ext groups in Serre quotients can be computed via direct limits of Ext groups in the ambient category.

The isomorphism for Ext^1 requires the subcategory to be localizing.

Higher Ext groups require additional assumptions on the subcategory.

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 $\mathrm{Ext}$.