exterior algebras

Let $A$ be a commutative algebra. Let $M$ be an $A$ -module.

The tensor algebra $\otimes_A M$ is defined as

$$\otimes_A M= A \oplus (\bigoplus_{j=1}^\infty M^{\otimes j}).$$

It is a (non commutative) $\mathbb{N}$ -graded algebra ¹ over $A$ . Let us define the following two sided ideal of $\otimes_A M$ .

$$I_{\text{symmetric}}=(x\otimes y -y\otimes x ; x\in M,y\in M).$$

$$I_{\text{exterior}}=(x\otimes x ; x\in M).$$

Then we define

$$S_A M=(\otimes_A M)/I_{\operatorname{symmetric}}$$

$$\wedge_A M=(\otimes_A M)/I_{\operatorname{exterior}}$$