Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that the restriction of $F$ to the special fiber is torsion free?



Torsion free sheaves in flat families
Not in general. For example, take $X=\mathbb{P}^1\_R$ and let $x\in X$ be a closed point on the special fiber. Then $I\_x$, the ideal sheaf of $x$ is torsion free (and thus flat over $R$), but the restriction to the special fiber is not torsion free.


