The factor sheaf of coherent sheaf

Question

If $\mathcal F$ is a coherent sheaf on a Noetherian scheme $X$ and $\mathcal F_1$ is a subsheaf of $\mathcal F$, then is it necessarily that the factor sheaf $\mathcal{F/F_1}$ on $X$ a coherent sheaf as well?

Answer 1

It is a general statement that on a Noetherian scheme $X$, in a short exact sequence of sheaves of $\mathcal O_X$-modules, coherence of any two of the sheaves in the s.e.s. implies coherence of the third. So, in the notation of your question, the quotient $\mathcal F/\mathcal F_1$ will be coherent if and only if $\mathcal F_1$ is coherent.

Now $\mathcal F_1$ need not be coherent. E.g. if $X$ is is irreducible but not zero-dimensional and $x$ is any closed point of $X$, and if $j: U = X \setminus \{x\} \to X$ is the indicated open immersion, then $j_! \mathcal O_X$ embeds naturally as an ideal sheaf of $\mathcal O_X$, with quotient equal to the local ring $\mathcal O_{X,x}$ supported just as the closed point $x$. Neither $j_!\mathcal O_X$ or $\mathcal O_{X,x}$ supported at $x$ are coherent. (One way to see this is that the closed point $x$, equipped with the sheaf of rings $\mathcal O_{X,x}$, is not a scheme.)