Exercise 55, § 1

Let C be a ring. Let G be a monoid and let $C^{(G)}$ be its algebra over C. Instead of $\mathrm{Rep}_C(C^{(G)})$ and $R_C(C^{(G)})$, we shall write $\mathrm{Rep}_C(G)$ and $R_C(G)$. If $\alpha : G' \to G$ is a homomorphism of monoids, we shall also denote by $\alpha$ the homomorphism of C-algebras $C^{(G')} \to C^{(G)}$ which it defines, and by $\alpha_* : R_C(G) \to R_C(G')$ the corresponding homomorphism.

By A, VIII, § 10, No. 5, there exists on $R_C(G)$ a structure of (commutative) ring such that

if E and F are modules of type $\mathrm{Rep}_C(G)$. The identity element for this multiplication is the class of the module $C_1$, equal to C with trivial action of G.

a) The ring $R_C(G)$ admits a unique structure of pre-$\lambda$-ring such that

for every module E of type $\mathrm{Rep}_C(G)$. (Observe first of all that $\Lambda^n(E)$ is again a module of type $\mathrm{Rep}_C(G)$. Next, if F is a submodule such that F and E/F are of type $\mathrm{Rep}_C(G)$, now prove that

in $R_C(G)$. For this purpose let us denote by $L_p$ the image of $\Lambda^p(F) \otimes_C \Lambda^{n-p}(E)$, under multiplication, in $\Lambda^n(E)$. We have $\Lambda^n(E) = L_0 \supset L_1 \supset ... \supset L_n = \Lambda^n(F) \supset L_{n+1} = 0$, and there exists a canonical isomorphism of $C^{(G)}$-modules from $L_p/L_{p+1}$ onto $\Lambda^p(F) \otimes \Lambda^{n-p}(E/F)$.

b) For every homomorphism $\alpha : G' \to G$ of monoids, the map $\alpha_* : R_C(G) \to R_C(G')$ is a $\lambda$-morphism. In particular $K_0(C)$ is a pre-$\lambda$-ring and $\varepsilon_* : R_C(G) \to K_0(C)$ is a $\lambda$-morphism. The homomorphism $G \to \{1\}$ furnishes a right inverse for it.

c) Let G be a monoid and R a set. A function $f : G \to R$ will be said to be central if $f(st) = f(ts)$ for every $s, t \in G$. Let us denote by FC(G, R) the set of these functions. If R is a $\lambda$-ring, FC(G, R) is canonically endowed with a structure of $\lambda$-ring such that

for every $f, f'$ in FC(G, R) and s in G. This applies in particular when $R = \Lambda(C)$.

d) Let E be a module of type Rep_C(G). For every s ∈ G, let us denote by s_E the homothety of ratio s in E, and put

(cf. p. 62, Exercise 49). Now prove that

defines a $\lambda$-morphism

e) If C is a field, then $\overline{\chi}$ is injective (A, VIII, § 10, No. 6, Proposition 10). Deduce in this case that $R_C(G)$ is a $\lambda$-ring.