Let $\mathcal{O}$ be a discrete valuation ring with finite residue field $k$ of characteristic $p$.
Let $S=\mathrm{Spec}(R)$ be a noetherian scheme over $\mathrm{Spec}(\mathcal{O})$. Are there sufficient conditions on $S[1/p]$ and $S \mod p$ so that $S$ is irreducible ?