Behavior of canonical divisors under purely inseparable base changes

Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X{\times }_k{k}^{1\mathrm{/}{p}^{\mathrm{\infty }}}{)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$. In this paper, we show that $K_Y+C=f^{\ast }{K}_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field.