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 $\mathcal{ A}$.
Categories: math.CT