pairing of exterior algebras

Let $A$ be a commutative ring. Let $M,N$ be $A$ -modules. Assume there is a $A$ -bi-linear pairing

$$\langle \bullet, \bullet \rangle : M \times N \to A.$$

Then we define a pairing of tensor algebras

$$\langle \bullet, \bullet \rangle_{\wedge} : \otimes_A M \times \otimes_A N \to A$$

defined by the following equation.

$$\langle v_1\otimes v_2\otimes \dots \otimes v_s, w_1\otimes w_2\o... ...w_j\rangle)_{i j} ) & (\text{if }s=t) \\ 0 & (\text {if }s\neq t ) \end{cases}$$

(The reader may prefer the expression

$$\langle v_1\otimes v_2\otimes \dots \otimes v_s, w_1\otimes w_2\otimes \dots \otimes w_s \rangle_\wedge$$

$$= \sum_{\sigma \in \mathfrak{S}_s} \operatorname{sgn}(\sigma) \lang... ...\dots \langle v_{s-1}, w_{\sigma(s-1)}\rangle \langle v_s, w_{\sigma(s)}\rangle$$

rather than the expression above which uses a determinant.)

Then it is easy to see that the pairing above descends to define a pairing of exterior algebras

$$\langle \bullet, \bullet \rangle_{\wedge} : \wedge_A M \times \wedge_A N \to A.$$