$(0\leq i\leq r)$ should read $(0\leq i\leq r-1)$.
This line should read: “We define the sub-DGA $N\mathscr{P}^{(m),\bullet}_{X/S}$ of $\mathscr{P}^{(m),\bullet}_{X/S}$ by”.
$\cap^{r+1}_{i=0}$ should read $\cap_{i=0}^{r-1}$.
$\cap^{r+2}_{i=0}$ should read $\cap_{i=0}^r$.
“a zero morphism” should read “a zero morphism at degree $\geq 1$“.