# Torsion free quasi-coherent sheaves

Let $R$ be an integral domain and $M$ be a torsion free injective module. Then $M$ is of the form $\oplus_{I}K$ (for some index set $I$), where $K$ is the quotient field of $R$.

Now let $\cal F$ be a torsion free injective quasi coherent sheaf over an integral scheme, and ${\cal K}_X$ denote the quasi-coherent sheaf determined by $U\mapsto \rm Frac({\cal O}_X(U))$. Then :

Is any torsion free qusai-coherent injective sheaf of the form $\oplus_{I}{\cal K}_X$ for some index set $I$?

This should be a consequence of classification of injective quasi-coherent $$\mathcal{O}_X$$-modules over (locally) Noetherian schemes:
By Theorem II.7.17 of Hartshorne's Residues and Duality (cf. B. Conrad's Grothendieck Duality and Base change, Lemma 2.1.5), every injective quasi-coherent sheaf on $$X$$ is of the form $$\bigoplus_i \mathscr{J}(x_i),$$ for some collection of (possibly repeating) points $$\{x_i \;|\; i \in I\}$$, where $$\mathscr{J}(x_i)=\nu_*(\widetilde{ E(k(x_i))}),$$ $$\nu: \mathrm{Spec}\,\mathcal{O}_{X, x_i} \rightarrow X$$ is the canonical map and $$E(k(x_i))$$ is the $$\mathcal{O}_{X, x_i}$$-injective hull of the residue field $$k(x_i)$$.
In the case when $$X$$ is integral, the torsion-free assumption clearly excludes contribution of any point $$x_i \in X$$ apart from the generic point (note that $$\mathscr{J}(x_i)$$ has support $$\overline{\{x_i\}}$$).
Thus, a torsion-free injective quasi-coherent sheaf on an integral scheme is of the form $$\mathscr{J}(\eta)^{\oplus I}$$ for some set $$I$$, where $$\eta$$ is the generic point of $$X$$. But it is easy to see that $$\mathscr{J}(\eta)=\nu_*(\widetilde{E(k(\eta))})=\nu_*(\widetilde{\mathcal{O}_{X, \eta}})=\mathcal{K}_X$$.