Exercise 56, § 1

Let $G$ be the free monoid generated by an element T, so that $C^{(G)}$ is identified with $C[T]$. Let $C_0$ denote the module $C[T]/TC[T]$. Then $[C_0]$ is an idempotent element of $R_C(C[T])$. The kernel of the canonical $\lambda$-morphism $R_C(C[T]) \to K_0(C)$ is the ideal generated by $1 - [C_0]$. Put

a) Now prove that $[C_0].R_C(C[T])$ is a $\lambda$-ideal, hence that $\tilde{R}_C(C[T])$ admits a quotient structure of pre-$\lambda$-ring. b) For every module E of type $Rep_C(C[T])$, let $T_E$ denote the homothety of ratio T in E. We have

and

is the characteristic polynomial of $T_E$ (cf. exerc. 49, p. 62). Now prove that $[E] \mapsto \overline{\chi}_E(T)$ defines a $\lambda$-morphism $R_C(C[T]) \to \Lambda(C)$ whose kernel contains $[C_0]$. By passing to the quotient, one obtains a $\lambda$-morphism

c) Let $\Lambda_{rat}(C)$ denote the subgroup of $\Lambda(C)$ generated by the elements of $\Lambda(C)$ which are polynomials in T. Now prove that the image of $\overline{\chi}$ is $\Lambda_{rat}(C)$. Conclude that $\Lambda_{rat}(C)$ is a sub-$\lambda$-ring of $\Lambda(C)$.

d) For every $r \in \mathbf{Z}$, define $\sigma_r : C[T, T^{-1}] \to C[T, T^{-1}]$ by $(\sigma_r f)(T) = T^r f(( - T)^{-1})$. We have $(\sigma_r(\sigma_s f))(T) = (-1)^s T^{r-s} f(T)$, in particular $\sigma_r(\sigma_r f) = (-1)^r f$, and $(\sigma_r f).(\sigma_s g) = \sigma_{r+s}(f.g)$, whatever $r, s$ in $\mathbf{Z}$ and $f, g$ in $C[T, T^{-1}]$. If $f$ is a polynomial in T of degree $\leq r$ ($r \geq 0$), then $\sigma_r f$ is one also. If moreover $f \in \Lambda(C)$ (i.e. $f(0) = 1$), then $\sigma_r f$ is monic of degree $r$. If E is a $C[T]$-module which is a free $C$-module of rank $r$, then $\sigma_r(\overline{\chi}_E(T)) = \chi_E(T)$.

e) If $f \in \Lambda(C)$ is a polynomial in T of degree $\leq r$, put $E_{r,f} = C[T]/\sigma_r f.C[T]$; it is a free $C$-module of rank $r$. If $f = 1$ we have $E_{r,1} = C[T]/T^r C[T]$. If $g \in \Lambda(C)$ is a polynomial in T of degree $\leq s$, we have an exact sequence of $C[T]$-modules

(use d)). Now prove that the class $\tau(f)$ of $E_{r,f}$ in $\tilde{R}_C(C[T])$ is independent of $r$ ($\geq \deg(f)$). (Indeed $\sigma_{r+s} f = (\sigma_s 1).(\sigma_r f) = T^s.(\sigma_r f)$, and the class of the module $C[T]/T^s C[T]$ in $\tilde{R}_C(C[T])$ is zero.) Now prove that if $g$ is a polynomial in $\Lambda(C)$, one has $\tau(fg) = \tau(f) + \tau(g)$. (Use again the exact sequence (*).) Deduce from this that $\tau$ extends to a group homomorphism

f) Now prove that $\overline{\chi} \circ \tau$ is the identity mapping of $\Lambda_{rat}(C)$. (Use the fact that if $f \in \Lambda(C)$ is a polynomial of degree $\leq r$, then the characteristic polynomial of $T_{E_{r,f}}$ is $\sigma_r f$.) g) Now prove that every element of $\tilde{R}_C(C[T])$ is the class of a $C[T]$-module E which is a free $C$-module.

h) Let E be a C[T]-module, and denote by $u \in \mathrm{End}_C(E)$ the homothety of ratio T in E; let $\overline{u} = \mathrm{Id}_{C[T]} \otimes_C u \in \mathrm{End}_{C[T]}(E[T])$. We have an exact sequence of C[T]-modules

(A, III, p. 106). Suppose that E is a free C-module with basis $(e_1, ..., e_r)$. Then E[T] is a free C[T]-module with basis $(1 \otimes e_i)$. If the matrix of $u$ for the basis $(e_i)$ is $(a_{ij})$, the matrix of $T - \overline{u}$ for the basis $(1 \otimes e_i)$ is $(T \delta_{ij} - a_{ij})$.

i) Let $f = (f_{ij})_{1 \leq i,j \leq r}$ be a matrix with coefficients in C[T]. We shall say that $f$ is special if, whatever $i, j$ in $\{1, ..., r\}$, $i \neq j$, $f_{ii}$ is a monic polynomial and $\deg(f_{ii}) > \deg(f_{ij})$. Now prove that $\det(f)$ is a monic polynomial (of degree $\deg(f_{11}) + \cdots + \deg(f_{rr})$) and that $f$ defines an injective endomorphism of C[T]''. Deduce from h) that every C[T]-module which is a free C-module of rank $r$ is isomorphic to the cokernel of such an endomorphism of C[T]''.

j) Let $f$ be a special matrix as in i). Put $f = \begin{pmatrix} f_{11} & f_+ \\ f_- & f' \end{pmatrix}$ where $f_+, f_-, f'$ are matrices of types $(1, r-1)$, $(r-1, 1)$ and $(r-1, r-1)$ respectively. Put

where $\overline{f} = f_{11} f' - f_- f_+$. Now prove that $h$ is a special matrix. Moreover, by identifying square matrices with the corresponding endomorphisms, one obtains exact sequences of C[T]-modules (where $\mathrm{Coker}(g)$ is isomorphic to $\mathbf{C}[T]^r / f_{11} \mathbf{C}[T]^{r-1}$) and Deduce from this, by induction on $r$, that $\mathrm{Coker}(f)$ is a projective C-module whose class in $R_C(C[T])$ is a $\mathbf{Z}$-linear combination of elements of the form $[C[T]/pC[T]]$ where $p$ is a monic polynomial.

k) Use the preceding parts f), g), i) and j) to show that $\tau : \Lambda_{rat}(C) \to \tilde{R}_C(C[T])$ is surjective, hence that $\overline{\chi} : \tilde{R}_C(C[T]) \to \Lambda_{rat}(C)$ is an isomorphism.

l) Now prove that $\Lambda_{rat}(C)$ is stable by $V_n$ and $F_n$ for each $n \in J$. There correspond therefore to them endomorphisms $V_n$ and $F_n$ of $\tilde{R}_C(C[T])$, by the preceding isomorphism. Let $\varphi_n$ be the homomorphism of C-algebras of C[T] into itself which maps T onto $T^n$. Now prove that $V_n$ (resp. $F_n$) is deduced from the endomorphism $\varphi_n^*$ (resp. $(\varphi_n)_*$) of $R_C(C[T])$ by passing to quotients.

In the exercises below, $p$ is a fixed prime number, and the Witt vector rings are relative to this prime number.