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Relations of derivations.

LEMMA 5.5   Let $ A$ be a ring. Let $ B$ be a commutative $ A$ -algebra. For any $ x \in \operatorname{Der}_A(B)$ , we have

$\displaystyle [L_x, d]= 0
$

PROOF.. This is based on the fact that $ d$ is naturally defined. A direct proof is obtained by using Lemma 5.4 and by noting the following facts.
  1. $\displaystyle (L_x d) f= L_x (df )=d (x. f)= d L_x f
$

  2. $\displaystyle (L_x d) d g =0 , \qquad
(d L_x) d g =d (d L_x(g))=0
$

$ \qedsymbol$

LEMMA 5.6   Let $ A$ be a commutative ring. Let $ B$ be a commutative $ B$ -algebra.
  1. For any $ x \in \operatorname{Der}_A(B)$ , we have

    $\displaystyle [d, i_x]_+ =L_x.
$

  2. For any $ x,y \in \operatorname{Der}_A(B)$ , we have

    $\displaystyle [L_x,i_y]=i_{[x,y]}.
$

PROOF.. Same method as above works. $ \qedsymbol$



2012-02-29