Definition 1

Let $A$ be a commutative ring and $M$ an $A$-module. The exterior algebra of $M$, denoted by $\bigwedge(M)$ or $\mathrm{Alt}(M)$ or $\bigwedge_A(M)$, is the algebra over $A$ the quotient of the tensor algebra $T(M)$ by the two-sided ideal $\mathfrak{J}''$ (also denoted by $\mathfrak{J}_{M}''$) generated by the elements $x \otimes x$, where $x$ runs through $M$.

Since the ideal $\mathfrak{J}''$ is generated by homogeneous elements of degree 2, it is a graded ideal (II, § 11, no. 3, Proposition 2); we write $\mathfrak{J}_n'' = \mathfrak{J}'' \cap T^n(M)$; the algebra $\bigwedge(M)$ is therefore graded by the graduation (called canonical) consisting of the $\bigwedge^n(M) = T^n(M)/\mathfrak{J}_n''$. Then $\mathfrak{J}_0'' = \mathfrak{J}_1'' = \{0\}$ and hence $\bigwedge^0(M)$ is identified with $A$ and $\bigwedge^1(M)$ with $T^1(M) = M$; in what follows we shall always make these identification and the canonical injection $M \to \bigwedge(M)$ will be denoted by $\phi''$ or $\phi_M''$.