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

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 $\mathcal{A}$ / $\mathrm{k}\mathrm{e}\mathrm{r}$ \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 $\mathcal{B}=\mathfrak{C}\mathfrak{o}\mathfrak{h}X$ of coherent sheaves on $X$. This gives a direct proof that $\mathfrak{C}\mathfrak{o}\mathfrak{h}X$ is a Serre quotient of $\mathcal{A}$.