appendix

In this subsection we prove a formula which play a fairly important role in our theory.

PROPOSITION 6.4   Let $p$ be a prime number. Let $D$ be a derivation on a commutative algebra $C$ of characteristic $p$ . Assume that there exists a non commutative algebra $A$ which contains $C$ as a subalgebra and that there exists an element $x\in A$ such that

$$[x, f]=D(f)$$

holds for all $f\in C$ . Then for any element $f$ of $C$ we have

$$(x+f)^p=x^p+D^{p-1}(f)+f^p$$

PROOF.. We substitute $a=f$ and $b=x$ in the Proposition 6.2. We need to know $\operatorname{ad}(T f + x)^{p-1} f$ . To do that, first we see by induction that

$$\operatorname{ad}(T f + x)^k f=D^k f$$

holds for any $k\in \mathbb{N}$ . In particular,

$$\operatorname{ad}(T f + x)^{p-1} f=D^{p-1} f$$

So

$$s_j(f,x)=(1/j) \operatorname{coeff}(\operatorname{ad}(T f+x)... ... D^{p-1} f & \text{if } j=1\\ 0 &\text{otherwise}. \end{cases}$$

$\qedsymbol$