\det\mathcal{E}\to \mathcal{E}\to \mathcal{F}\to 0$$ with $\mathcal{F}$ a vector bundle. Since $\det\mathcal{F}\cong \mathcal{O}$, the assumption implies that $\mathcal{F}\cong \mathcal{O}^{\oplus r-1}$. Since $H^1(\det\mathcal{E})=0$, this implies that $\mathcal{E}\cong \mathcal{O}^{\oplus r-1}\oplus\det\mathcal{E}$.

The assumption of Proposition \[dsubExist\] is satisfied if $X$ is either a projective space $\mathbb{P}^n$ or a hyperquadric $\mathbb{Q}^n$. See, e.g., [@oss Chap. 1 Theorem 3.2.1] and [@w3 Lemma 3.6.1]

\[quotientTorsionFree\] Let $\mathcal{F}$ be a locally free coherent sheaf, $\mathcal{G}$ a torsion-free coherent sheaf, and let $$0\to \mathcal{F}\to \mathcal{G}\to \mathcal{H}\to 0$$ be an exact sequence of coherent sheaves on $X$. If the support $Z$ of torsion subsheaf $\mathcal{T}$ of $\mathcal{H}$ has codimension $\geqq 2$ in $X$, then $\mathcal{T}=0$, i.e., $\mathcal{H}$ is torsion-free.

Set $U=X\setminus Z$, and let $i:U\to X$ be the inclusion. Let $\varphi:\mathcal{G}\to \mathcal{H}/\mathcal{T}$ be the composite of the two quotients $\mathcal{G}\to \mathcal{H}$ and $\mathcal{H}\to \mathcal{H}/\mathcal{T}$. Let $\mathcal{K}$ be the kernel of $\varphi$. We have the following exact sequence by the snake lemma. $$0\to \mathcal{F}\xrightarrow{\psi} \mathcal{K}\to \mathcal{T}\to 0$$ Since the support of $\mathcal{T}$ is outside $U$, we see that $\mathcal{F}|_U\cong \mathcal{K}|_U$. Since $\mathcal{K}$ is a subsheaf of a torsion-free sheaf $\mathcal{G}$, $\mathcal{K}$ is torsion-free. Hence the canonical morphism $\mathcal{K}\to i_*(\mathcal{K}|_U)$ is injective. On the other hand, we have isomorphisms $\mathcal{F}\cong i_*(\mathcal{F}|_U)\cong i_*(\mathcal{K}|_U)$. 