Homological dimension of pure coherent sheaves and specialization

Question

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.

Answer 1

Take for $X$ the plane curve $X^3=Y^2T$, and for $F$ the ideal sheaf $(X-\pi ^2T,Y-\pi ^3T)$. Its restriction to the generic fiber is the ideal of a smooth point, hence is an invertible sheaf, while its restriction to the special fiber is the ideal of the singular point $(0,0,1)$, which has infinite homological dimension.