Therefore $\psi$ is an isomorphism, and thus $\mathcal{T}\cong 0$. Hence $\mathcal{H}$ is torsion-free.

Recall here that $\det:K(X)\to {\mathop{\rm Pic}\nolimits}X$ is defined since $X$ is smooth and projective and thus every coherent sheaf admits a finite locally free resolution. Here $K(X)$ denotes the Grothendieck group of $X$.

In the rest of this section, we assume that ${\mathop{\rm Pic}\nolimits}X\cong \mathbb{Z}$, and let $\mathcal{O}_X(1)$ denote the ample generator of ${\mathop{\rm Pic}\nolimits}X$. Let $d$ be the integer such that $\det\mathcal{E}\cong \mathcal{O}_X(d)$.

\[nefquotient\] If $\mathcal{G}$ is a quotient coherent sheaf of $\mathcal{E}$, then $\det\mathcal{G}$ is nef. Moreover if $\det\mathcal{G}\cong \mathcal{O}_X$ then the support of the torsion subsheaf of $\mathcal{G}$ has codimension $\geqq 2$ in $X$.

First suppose that $\mathcal{G}$ is torsion free, and let $Z$ be the singular locus of $\mathcal{G}$, i.e., the locus where $\mathcal{G}$ is not locally free. Then $Z$ has codimension $\geqq 2$. Set $U=X\setminus Z$, and let $i:U\to X$ be the inclusion. Observe that $\det \mathcal{G}$ is equal to the sheaf $i_*(\det(\mathcal{G}|_U))$. Let $s$ be the rank of $\mathcal{G}$. Then the surjection $\mathcal{E}\to \mathcal{G}$ induces a morphism $\wedge^{s}\mathcal{E}\to \det\mathcal{G}$ which is surjective on $U$. Suppose, to the contrary, that $\det\mathcal{G}$ is not nef. Then $\det\mathcal{G}$ is isomorphic to $\mathcal{O}(k)$ for some 