Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined by exactly $d$ homogeneous polynomials and no set of polynomials of cardinality less than $d$ has $X$ as its zero set. Suppose that $f_1$,..., $f_d$ is a set of polynomials defining $X$. If I lift these polynomials naively to the $p$-adic integers $\mathbb{Z}_p$ (call the lifts $F_1$,..., $F_d$), I will get a projective algebraic set $X' \subseteq \mathbb{P}^n_{\mathbb{Z}_p}$.
My question is this: is $X'$ flat over $\mathbb{Z}_p$? Put differently, is the module $\mathbb{Z}_p[x_0,...x_n]/<F_1, ..., F_d>$ flat over $\mathbb{Z}_p$? I know that $\mathbb{Z}_p$ is a DVR, so flatness is equivalent to being torsion-free, but I just cannot see how to prove that it is flat (assuming that that is true at all). In the case when $X$ is smooth, $X'$ is easily seen to be smooth, hence flat.