interior derivation

We employ the same assumption of the previous subsection. For any $x\in N$ , we define

$$i_x: \wedge_A M \to \wedge_A M$$

to be the unique $A$ -linear map which satisfies the following properties.

1. $i_x(f)=0$ for any $f \in A$ .
2. $i_x(m)=\langle m,x\rangle$ for any $m\in M$ .
3. $i_x$ is an odd derivation. That means,
   $$i_x(\alpha \wedge \beta) =(i_x\alpha)\wedge \beta +(-1)^s \alpha ... ...beta) \quad (\forall \alpha \in \wedge^s_A M, \forall \beta \in \wedge_A M).$$

The proof of the well-definedness of $i_x$ is similar to the proof of well-definedness of $d$ . We leave the detail to the reader.

Using the above definition we may also explicitly write down a formula for the interior derivations.

$$i_x(m_1\wedge m_2 \wedge \dots \wedge m_s)= \sum_{j=1}^s (-1)^j \... ...2 \wedge \dots\wedge \overset{\text{omit}}{\hat{m_j}}\wedge \dots \wedge m_s).$$

This leads to an important adjunction relation

$$\langle i_x \alpha, \beta \rangle_\wedge =\langle \alpha, x\wedge... ...ngle_\wedge \qquad (\forall \alpha \in \wedge M, \forall \beta \in \wedge N).$$

(We may verify the above equation by using the ``determinant expansions by minors''.)

In particular, we note that for any $x,y\in N$ , an anti commutation relation

$$[i_x, i_y]_+=0$$

holds. Another useful equation is

$$i_{x_s} i_{x_{s-1}}\dots i_{x_2} i_{x_1} \alpha =\langle \alpha,x... ...e} \qquad (\forall x_1,x_2,\dots,x_s \in N, \forall \alpha \in \wedge^s_A M).$$