Proposition 16

— Let E be a finite dimensional vector space over a commutative field K, let $\mathfrak{F}$ be a set of endomorphisms of E and let A be the subalgebra of $\mathrm{End}_K(E)$ generated by $\mathfrak{F}$ and Id. Then the following conditions are equivalent :

(i) There exists an extension L of K such that $\mathfrak{F}_{(l.)}$ is diagonalisable.

(ii) The K-algebra $A$ is étale (V, p. 28, Def. 1).

(iii) The elements of $\mathfrak{F}$ are absolutely semi-simple and commute with one another.

Note first that, for every extension $L$ of $K$, the $L$-algebra generated by $\mathfrak{F}_{(L)}$ and $\mathrm{Id}_{E_{(L)}}$ coincides with $L \otimes_K A$; hence by Prop. 13 $\mathfrak{F}_{(L)}$ is diagonalisable if and only if the $L$-algebra $L \otimes_K A$ is diagonalisable. The equivalence of conditions (i) and (ii) thus follows from V, p. 28, Def. 1. On the other hand, it is immediate that (i) $\Rightarrow$ (iii). Finally suppose (iii) holds, and let $L$ be an algebraic closure of $K$; then the elements of $\mathfrak{F}_{(L)}$ are diagonalisable (VII, p. 40, Prop. 12) and commute with one another; hence $\mathfrak{F}_{(L)}$ is diagonalisable by VII, p. 41, Prop. 13.