I am studying N. [Katz's paper][1] where I found a fairly good account on $ p$ -curvatures.

I don't understand the following proof:

[![enter image description here][2]][2]

Let : 

$\Psi: Der(S|T) \to End_T(\mathcal E)$   

$D \to (\nabla(D))^p -\nabla(D^p)$

Where $\nabla: Der(S|T) \to End_T(\mathcal E)$ such that: $\nabla(D)(ge) = D(g)e+g\nabla(D)(e)$, $g$ and $D$ sections of $\mathcal O_S$ and $Der(S|T)$ respectively.

 1. To proof $(5.4.4)$, we have by additivity of the p-curvature:

 $\Psi(D)=\sum_i a_i^p \Psi( \frac{\partial}{\partial s_i}) = \sum_i a_i^p (\nabla(\frac{\partial}{\partial^p s_i}))^p -\sum_i a_i^p \nabla(\frac{\partial^p}{\partial s_i^p})$ but the term $\sum_i a_i^p \nabla(\frac{\partial^p}{\partial s_i^p})$ disappears in the proof and I don't see why?

 2. At the end of the proof, it looks like we use the fact that $\frac{\partial}{\partial s_i}$ and $\frac{\partial}{\partial s_j}$ commute, but why is that true?

Thank you for your help.

  [1]: http://www.numdam.org/article/PMIHES_1970__39__175_0.pdf
  [2]: https://i.sstatic.net/Z0xcn.png