I'm looking for a proof (or better, a reference) for the following claim.
Claim: Let $X$ be an (irreducible) variety defined over $\mathbb{Z}$. Let $\nu:\tilde{X}\rightarrow X$ be its normalization. Then, for large enough primes $p$, $\tilde{X}/p$ is the normalization of $X/p$.
The way I was trying to do this was by appealing to Serre's "normal = $R1+S2$" criterion. Doing this, it isn't difficult to prove that $X/p$ is $R1$ for all large primes. I'm having more difficulty with the $S2$ part.
So, I'd appreciate either an elementary/direct proof of the above claim or a proof (or counterexample) of "if $X$ is $S_k$ over $\mathbb{Z}$ then $X/p$ is $S_k$ over $\mathbb{F}_p$ for large enough primes $p$''.