Exercise 2, § 5

(Functor $\Gamma$) Denote by $k$ a commutative ring and by $E$ a $k$-module. By the gamma algebra of $E$ we understand the associative commutative unital algebra defined by the generating system $N \times E$ and by the relators (cf. III, p. 450)

where $p, q$ range over $\mathbf{N}$, $x, y$ range over $E$ and $\lambda$ over $k$. This algebra is denoted by $\Gamma(E)$. For each $p \in \mathbf{N}$ we denote by $\gamma_p$ the mapping of $E$ into $\Gamma(E)$ composed of the injection $x \mapsto (p, x)$ and the canonical homomorphism of the free commutative algebra on $N \times E$ into $\Gamma(E)$. a) Verify that for each $x \in E$, the sequence $\gamma_p(X)$ is a sequence of exponential type of elements of $\Gamma(E)$ (Exercise 1) and that the mapping $y : x \mapsto (\gamma_p(x))_{p \in \mathbf{N}}$ is a homomorphism of the $k$-module $E$ into the $k$-algebra $\mathscr{E}(\Gamma(E))$. b) (Universal property of $\Gamma(E)$). Let $B$ be an associative commutative unital $k$-algebra and $\varphi$ a homomorphism of the $k$-module $E$ into $\mathscr{E}(B)$. Show that there exists a unique algebra homomorphism $h : \Gamma(E) \to B$ such that $\varphi = \mathscr{E}(h) \circ y$. c) Show that there exists a unique unital homomorphism of algebras $\varepsilon : \Gamma(E) \to k$ such that $\varepsilon(\gamma_p(x)) = 0$ for all $p > 0$ and all $x \in E$. d) For each $p \in \mathbf{N}$ let $\Gamma_p(E)$ be the submodule of $\Gamma(E)$ generated by the elements

where $s \in \mathbf{N}$, $\nu \in \mathbf{N}^s$ and $|\nu| = \sum_i \nu_i = p$. Verify that the submodules $\Gamma_p(E)$ form a graduation of the algebra $\Gamma(E)$. Show that $\gamma_1$ is an isomorphism of $E$ onto $\Gamma_1(E)$ (to verify that $\gamma_1$ is injective, consider the algebra $B = k \times E$ with the product defined by the formula $(\lambda, x)(\mu, y) = (\lambda \mu, \lambda y + \mu x)$ and show that there exists a homomorphism of $\Gamma(E)$ into $B$ which maps $\gamma_1(x)$ to $(0, x)$ for any $x \in E$. e) Let $E$ and $F$ be two $k$-modules and $h$ a homomorphism of $E$ into $F$. Show that there exists a unique homomorphism of graded algebras $\Gamma(h) : \Gamma(E) \to \Gamma(F)$ such that $\gamma_p \circ h = \Gamma(h) \circ \gamma_p$ for all $p \in \mathbf{N}$. Verify that if $g$ and $h$ are homomorphisms of $k$-modules such that $g \circ h$ is defined, then $\Gamma(g \circ h) = \Gamma(g) \circ \Gamma(h)$.

f) Let h : E → F be a surjective homomorphism of k-modules. Show that Γ(h) is surjective and that its kernel is the ideal of Γ(E) generated by the elements γ_p(x) with p > 0 and x ∈ Ker(h).

g) Let E and F be two k-modules. Show that there exists a unique homomorphism φ of the algebra T(E × F) into the algebra Γ(E) ⊗_k Γ(F) such that

for any p ∈ N, x ∈ E and y ∈ F. Show that this homomorphism is an isomorphism compatible with the graduations.