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Torsion-free sheaf cohomology over discrete valuation rings

Is it true that $H^i(\mathcal{O}_X)$ is a torsion-free $R$-module for $i>0$ i.e., for $K=\mathrm{Frac}(R)$, is the natural morphism from $H^i(\mathcal{O}_X)$ to $H^i(\mathcal{O}_X) \otimes_R K$ injective …
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