Properties of the derived subgroup of an algebraic group

Write $A=O(G)$ and let $c_n:G^{2n}\to G$ send its arguments to a product of $n$ commutators. Put $I={\bigcap }_{n\ge 1}\ker (c_n^{\ast }:A\to A^{\otimes 2n})$ and $B=A\mathrm{/}I$. The induced maps from $B$ to the target rings are jointly injective. Their pairwise tensor products are jointly injective on $B{\otimes }_{k}B$: for a finite tensor expression, restrict to the finite-dimensional spans of its factors; joint injectivity supplies finitely many separating coefficient functionals on each span. For $f\in I$, every such tensor map kills $\mathrm{\Delta }(f)$ because composing multiplication with $c_n\times c_m$ is $c_{n+m}$. Hence $\mathrm{\Delta }(I)\subseteq I\otimes A+A\otimes I$. Inversion reverses a commutator product and replaces each commutator by the one with its two arguments exchanged, so $S(I)\subseteq I$; evaluation at the identity gives $\epsilon (I)=0$. Thus $I$ is a Hopf ideal. The group $\mathrm{Spec}B$ contains the commutator morphism, and any closed subgroup containing it contains every $c_n$, so its defining Hopf ideal lies in $I$. Therefore $DG=\mathrm{Spec}B$.

For any field extension $K\mathrm{/}k$, express an element of $A{\otimes }_{k}K$ as a finite sum ${\sum }_i{a}_i\otimes {\lambda }_i$ with the ${\lambda }_i$ linearly independent over $k$. All $c_n^{\ast }$ kill this element exactly when they kill every $a_i$, so the joint kernel after extension is $I{\otimes }_{k}K$. If $G$ is smooth, every target $(A^{\otimes 2n}){\otimes }_{k}K$ is reduced, and their jointly injected subalgebra $B{\otimes }_{k}K$ is reduced. Thus $DG$ is geometrically reduced and smooth by [F2]. If $G$ is connected, its geometric connectedness in [F2] makes each finite product $G^{2n}$ connected. For an idempotent $b\in B$, every $c_n^{\ast }(b)$ is an idempotent on the connected affine scheme $G^{2n}$, hence a scalar $0$ or $1$. Evaluation at the identity shows that all these scalars equal $\epsilon (b)$. Joint injectivity of the maps from $B$ then gives $b=\epsilon (b)$, so $B$ has no nontrivial idempotents and $DG$ is connected. This proves (b), without treating $c$ as a group homomorphism or identifying a scheme-generated closure with a union of underlying images.

For every $k$-algebra $R$ the commutator of any two $G(R)$-points lies in $(DG)(R)$, since $c$ factors through $DG$; hence $[G(R),G(R)]\subseteq (DG)(R)$. Under the additional hypotheses of (d), [F3] gives equality. If $DG\subseteq H\subseteq G$, the identity $gh{g}^{-1}=[g,h]h$ for $g\in G(R),h\in H(R)$ proves $H(R)$ stable under conjugation for every $R$, which is scheme-theoretic normality. In particular $DG$ is normal. This proves (c) and (d).

The quotient $G\mathrm{/}DG$ exists by [F5]. Its commutator is trivial after pullback along the faithfully flat product cover $G\times G\to (G\mathrm{/}DG)\times (G\mathrm{/}DG)$, because the commutator of $G$ lands in its kernel; faithfully flat descent therefore makes the quotient commutative. The independent-coefficient argument of step 2.1 also works with an arbitrary $k$-algebra in place of $K$, considered as a $k$-vector space. Thus the description by the $c_n$ commutes with such base changes. Every automorphism of $G_R$ preserves commutator products and their generated closed subgroup, so it preserves $(DG{)}_R$. Hence $DG$ is characteristic as well as closed and normal, proving (a).

If smooth connected solvable $G\mathrm{\ne }1$ satisfied $DG=G$, every term of its derived series would equal $G$, contradicting termination at $1$. Therefore $DG\mathrm{\ne }G$. The group $G$ is geometrically integral by [F2], and $DG$ is a nonempty proper closed subgroup; [F4] gives $\dim DG<\dim G$. This proves (e) and all clauses.