Residue trivialization for a degree n+1 projective hypersurface ($K_Y\cong {\mathcal{O}}_Y$)

For a smooth hypersurface $Y=\{f=0\}\subset {\mathbb{CP}}^n$ of degree $n+1$, let $E=\sum z_i{\partial }_{z_i}$ and $\Omega =d{z}_0\wedge \cdots \wedge d{z}_n$. The meromorphic form $({\iota }_E\Omega )/f$ descends to projective space because its scaling weights cancel and it is horizontal. Its Poincaré residue is a nowhere-zero section of $K_Y$. On the chart $z_j=1$, with remaining coordinates $t_1,\dots ,t_n$ in original-index order and $F=f{|}_{z_j=1}$, it is $(-1{)}^{j+r-1}(d{t}_1\wedge \cdots \widehat{d{t}_r}\cdots \wedge d{t}_n)/F_{t_r}$ wherever $F_{t_r}\ne 0$. Thus $K_Y\cong {\mathcal{O}}_Y$.