Proposition 6

Assume that $k$ is infinite and that condition (AC) is satisfied. Let $k'$ be an extension of $k$. Put $V'= V\otimes_kk',S'= S\otimes_kk'$. Let $r': S'\rightarrow$ End(V$'$) be the map obtained from $r$ by extension of scalars. Then

The first equality follows from Prop. 1. To prove the second, we can assume that $V = V^0(S)$ and so $V'{= V'}^0(S)$. Let $(s_1, . . . , s_m)$ be a basis of S and $(e_1, . . . , e_n)$ a basis of V. There exist polynomials $P_{ij}(X_1, . . . ,X_m)$ such that

for $1\leq j\leq n$ and $a_1, . . . , a_m\in k'$. By hypothesis, $r'(s)^n= 0$ for all $s\in S$, in other words $P_{ij}(a_1, . . . , a_m) = 0$ for $1\leq i, j\leq n$ and $a_1, . . . , a_m\in k$. Since $k$ is infinite, $P_{ij}= 0$. Consequently, every element of $r'(S')$ is nilpotent and $V'{= V'}^0(S')$.