On $F$-pure inversion of adjunction

We analyze adjunction and inversion of adjunction for the $F$-purity of divisor pairs in characteristic $p>0$. In this vein, we give a complete answer for principal divisors under $\mathbb{Q}$-Gorenstein assumptions but without divisibility restrictions on the index. We also give a detailed analysis relating the $F$-purity of the pairs $(R,\mathrm{\Delta }+D)$ and that of $(R_D,{\text{Diff}}_D(\mathrm{\Delta }))$ motivated by Kawakita’s log canonical inversion of adjunction via reduction to prime characteristic.