Let $X$ be a projective variety, not necessarily smooth, $R$ a DVR with residue field $k$ (assume char$(k)=0$). I am looking for examples of a pure coherent sheaf, say $F$, on $X_R:=X \times_k \mathrm{Spec}(R)$ such that its restriction to the generic fiber has finite homological dimension but the restriction to the special fiber has infinite homological dimension. Any text which covers such examples is very welcome.

P.S. Also assume that $X$ is Cohen-Macaulay.