On the Ext-computability of Serre quotient categories

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 $\mathrm{Ext}$.

updated bibliography and deleted remaining occurrences of "maximally"

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Cited by in corpus (7)

• Gauge Backgrounds and Zero-Mode Counting in F-Theory
• Using the internal language of toposes in algebraic geometry
• On monads of exact reflective localizations of Abelian categories
• Gabriel morphisms and the computability of Serre quotients with applications to coherent sheaves
• A constructive approach to Freyd categories
• A constructive approach to the module of twisted global sections on relative projective spaces
• Flabby and injective objects in toposes