Exercise 13, § 7

(a) Let M be a graded A-module of type $\mathbf{N}$. Show that there exists on the algebra $S(M)$ (resp. $\Lambda(M)$) one and only one graduation of type $\mathbf{N}$ under which $S(M)$ (resp. $\Lambda(M)$) is a graded A-algebra and which induces on M the given graduation. Show that if the homogeneous elements of even degree in M are all zero (in which case the graduation on M is called odd), the graded algebra $\Lambda(M)$ thus obtained is alternating (§ 4, no. 9, Definition 7).

(b) Given a graded A-module of type $\mathbf{N}$, consider the following universal mapping problem: $\Sigma$ is the species of anticommutative graded algebra structure (§ 4, no. 9, Definition 7), the morphisms are A-homomorphisms of graded algebras (§ 3, no. 1) and the $\alpha$-mappings are the graded homomorphisms of degree 0 of M into an anticommutative graded A-algebra. Let $M^-$ (resp. $M^+$) denote the graded submodule of M generated by the homogeneous elements of odd (resp. even) degree. Show that the graded algebra $G(M) = \Lambda(M^-) \otimes_A S(M^+)$ is anticommutative; a canonical injection $j$ of M into this algebra is defined by writing $j(x) = x \otimes 1$ if $x \in M^-$, $j(x) = 1 \otimes x$ if $x \in M^+$. Show that the graded algebra $G(M)$ and the injection $j$ constitute a solution of the above universal mapping problem.

is called the universal anticommutative algebra over the graded A-module M.

(c) Prove the analogues of Propositions 2, 3 and 4 for the universal anticommutative algebra. Consider the case where M admits a basis consisting of homogeneous elements: find explicitly in this case a basis of the universal anticommutative algebra.