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 $\mathscr{Q}:\mathcal{A} \to \mathcal{B}$. 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 $\mathcal{A}$.