Exercise 2, Appendix 2

Let D be a local ring; we denote by $D^*$ the group of invertible elements of D, by $D^*_{ab}$ the quotient of $D^*$ by its derived group, and by $\pi : D^*\rightarrow D^*_{ab}$ the canonical homomorphism. Let V be a free right D-module of finite dimension $n$. We denote by B(V) the set of bases of V and by Ω(V) the set of mappings $\omega : B(V)\rightarrow D^*_{ab}$ satisfying the following two conditions:

(i) For $\lambda_1, . . . , \lambda_n$ in $D^*$ and $(v_1, . . . , v_n)\in B(V)$, we have $\omega (v_1\lambda_1, . . . , v_n\lambda_n) =$ $\omega (v_1, . . . , v_n)\pi (\lambda_1. . . \lambda_n)$.

(ii) For $i\not=j$ and $\lambda \in D$, we have $\omega (v_1, . . . , v_i+v_j\lambda , . . . , v_n) =\omega (v_1, . . . , v_n)$. a) Let $e$ be an element of V and W a submodule of V supplementary to $eD$. Let $\varphi \in \Omega(W)$; prove that there exists a unique element $\omega$ of Ω(V) satisfying

for every basis $(w_1, . . . , w_{n-1})$ of W (adapt the proof of Proposition 1).

b) Deduce that given two elements $\omega$ and $\omega '$ of Ω(V), there exists a unique element $t$ of $D^*_{ab}$ such that we have $\omega '=t\omega$.

c) Let $u$ be an automorphism of V. Prove that there exists a unique element of $D^*_{ab}$, denoted by det $u$, such that we have $\omega (u(v_1), . . . , u(v_n)) =$ (det $u$)$\omega (v_1, . . . , v_n)$ for every basis $(v_1, . . . , v_n)$ of V and every element $\omega$ of Ω(V). The mapping $u\mapsto$ det $u$ from $\mathbf{G}\mathbf{L}(V)$ to $D^*_{ab}$ is a group homomorphism.

d) Take $V = D^n_d$, so that $\mathbf{G}\mathbf{L}(V)$ is identified with the group $\mathbf{G}\mathbf{L}_n(D)$ of invertible square matrices of order $n$. Prove that the mapping det$:\mathbf{G}\mathbf{L}_n(D)\rightarrow D^*_{ab}$ is the unique group homomorphism with value 1 on all matrices $B_{ij}(\lambda ) (i\not=j,\lambda \in D)$ and satisfying det(diag($\lambda_1, . . . , \lambda_n$)) $=\pi (\lambda_1. . . \lambda_n)$ for $\lambda_1, . . . , \lambda_n$ in $D^*$.

e) Adapt Examples 2 through 6 of No.4 to this setting.