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

TL;DR

The paper establishes a fundamental homomorphism theorem analog for Serre quotients in Abelian categories, especially when no section functor exists, with applications to coherent sheaves on schemes like projective and toric varieties.

Contribution

It proves that certain exact and essentially surjective functors are equivalent to Serre quotients even without section functors, and applies this to coherent sheaves on schemes.

Findings

Characterizes Serre quotients without section functors.

Applies results to coherent sheaves on projective and toric varieties.

Provides a direct proof of the Serre quotient structure of 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 $\mathscr{Q}:\mathcal{A} \to \mathcal{B}$. It states that $\mathscr{Q}$ is up to equivalence the Serre quotient $\mathcal{A} \to \mathcal{A} / \mathrm{ker} \mathscr{Q}$, 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 $\mathcal{A}$.