Characterizing Serre quotients with no section functor and applications to coherent sheaves

Abstract

We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories $\mathcal{Q}:\mathcal{A}\to \mathcal{B}$. It states that $\mathcal{Q}$ is up to equivalence the Serre quotient $\mathcal{A}\to \mathcal{A}/\ker \mathcal{Q}$, even in cases when the latter does not admit a section functor. For several classes of schemes $X$, including projective and toric varieties, this characterization applies to the sheafification functor from a certain category $\mathcal{A}$ of finitely presented graded modules to the category $\mathcal{B}=\mathfrak{Coh}X$ of coherent sheaves on $X$. This gives a direct proof that $\mathfrak{Coh}X$ is a Serre quotient of $\mathcal{A}$.

updated bibliography

References in corpus (3)

• An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization
• On monads of exact reflective localizations of Abelian categories
• Equivariant Primary Decomposition and Toric Sheaves

Cited by in corpus (6)

• Gauge Backgrounds and Zero-Mode Counting in F-Theory
• On the Ext-computability of Serre quotient categories
• 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 the module of twisted global sections on relative projective spaces
• On subdirect factors of a projective module and applications to system theory