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 $I$?


1 Answer 1


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 $\mathscr{J}(\eta)=\nu_*(\widetilde{E(k(\eta))})=\nu_*(\widetilde{\mathcal{O}_{X, \eta}})=\mathcal{K}_X$.


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