Let $X$ be a smooth variety over $\mathbb{C}$, and let $\omega \in \operatorname{Pic}(X)_\mathbb{R}$ be an ample class. I would like to know if any $\mu_\omega$-semistable sheaf $E \in \operatorname{Coh}(X)$ admits a Jordan-Hölder sequence, i.e. a finite sequence $$0 = E_0 \subset E_1 \subset \dots \subset E_{n-1} \subset E_n = E,$$ such that the factors $E_i / E_{i+1}$ are $\mu_\omega$-stable.

Lehn and Huybrechts state something similar, the claim the existence of a Jordan-Hölder sequence with stable factors, which, according to the same book, is a weaker notion than $\mu$-stability. Also, I don't understand the proof they give:

Proof. Any filtration of $E$ by semistable sheaves with reduced Hilbert polynomial $p(E)$ has a maximal refinement, whose factors are necessarily stable.

Why is that true, and what exactly is a maximal refinement? Or am I missing something different here?

This is what I tried:

If $E$ is not already stable, the set of proper subsheaves $E' \subsetneq E$ with $\mu(E') = \mu(E)$ is non-empty. Then we can mod out a *maximal* subsheaf of that kind (which exists by Noetherianity), and obtain $F = E / E'$.

**Question 1:** Why is $F$ torsion free?

Equivalently, one could ask why $E'$ is saturated. It would also be sufficient if the saturation $\tilde{E''}$ of any submodule $E'' \subset E$ had the same slope, but I don't think this is true: If $\tilde{E''} / E''$ is supported in dimension $1$, then $c_1(\tilde{E''} / E'') \cdot \omega > 0$, so $\mu(\tilde{E''}) < \mu(E'')$.

Now suppose $F$ is torsion free. Then $\mu(F) = \mu(E)$, and $F$ is in fact $\mu$-stable, because any proper nonzero quotient $F \to Q \to 0$ with $\mu(Q) = \mu(F) = \mu(E)$ violates the maximality of $E'$. So inductively we can construct a sequence $$ E = E^0 \supset E^1 \supset E^2 \supset \dots,$$ whose quotients $F^i = E^i / E^{i+1}$ are torsion-free, stable sheaves of slope $\mu(E)$. Because the quotients are torsion-free, the ranks of the $F^i$ are strictly decreasing, so the sequence has to stop.

**Question 2:** Is the rest of my reasoning correct?

A bit of context: I'm reading Bridgeland's *Stability conditions on K3 surfaces*, and in the proof of **Lemma 6.2**, he writes:

The only nontrivial part is to check that if a torsion-free $\mu_\omega$-semistable sheaf $E$ satisfies $(\Delta − r\beta)\cdot\omega = 0$, then $Z(E) \in \mathbb R_{ >0}$. It is enough to check this when $E$ is $\mu_\omega$-stable.

I figured that a Jordan–Hölder sequence might prove this reduction step, and I don't see any other way to do it.