Nilpotency of locally isotropic $\mathrm{K}_1$-functor

Abstract: We show that the $\mathrm{K}<em>1$-functor modeled on locally isotropic reductive groups is hypoabelian if the base ring has finite Bass--Serre dimension and, for the Tits index $\mathsf{E}</em>{8, 2}^{78}$, contains a field. For classical or globally isotropic reductive groups schemes $\mathrm{K}_1$ is actually solvable. This implies that the elementary subgroup (or its derived subgroup) is the maximal perfect subgroup of the reductive group.

• The paper proves that K₁^G(K) is hypoabelian under finite Bass–Serre dimension and local isotropic rank at least 2, with [E_G(K), E_G(K)] as the largest perfect subgroup.
• A localization–completion argument establishes the commutator estimate [G⁰(K), Gᵈ(K)] ≤ Gᵈ⁺¹(K), extending Bak’s dimension-filtration method from linear groups to reductive group schemes.
• The results further show solvability for globally isotropic or classical groups, while highlighting unresolved cases such as the exceptional E₈ index, rank-one isotropy, and relative locally isotropic K₁-theory.

The paper establishes a nilpotency (hypoabelian and, in many cases, solvable) structure theorem for the $\mathrm{K}_1$-functor of reductive group schemes satisfying only a local isotropy condition. The result extends Bak's classical solvability theorem for the relative $\mathrm{K}_1$-functor of the general linear group from the linear setting to arbitrary reductive group schemes, and yields the conclusion that the derived subgroup of the elementary subgroup is the maximal perfect subgroup of the group of $K$-points.

Background and motivation

For a ring $R$ with finite Bass–Serre dimension, Bak proved that the relative $\mathrm{K}_1$-functor $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$ is solvable, via an explicit dimension filtration on $\mathrm{GL}(n, R, I)$ [(2608.17488), Introduction]. Analogous results were obtained for Chevalley groups and unitary groups. The present work addresses the case of reductive group schemes $G$ over a commutative ring $K$ that are isotropic of rank at least $2$ at every prime, but need not possess a globally defined proper parabolic subgroup. The elementary subgroup $\mathrm{K}_1$0 is defined functorially via colocalizations $\mathrm{K}_1$1 attached to a Zariski cover by isotropic pinnings, following the author's earlier work. The main theorem states:

If $\mathrm{K}_1$2, the local isotropic rank of $\mathrm{K}_1$3 is at least $\mathrm{K}_1$4, and $\mathrm{K}_1$5 contains a field whenever the Tits index $\mathrm{K}_1$6 appears in a maximal isotropic pinning, then $\mathrm{K}_1$7 is hypoabelian, i.e. $\mathrm{K}_1$8 is the largest perfect subgroup of $\mathrm{K}_1$9. Moreover, $K$0 is solvable if $K$1 either has an isotropic pinning of rank at least $K$2, or is simple of classical absolute type (avoiding triality for type $K$3).

Bass–Serre dimension

The paper works with the Bass–Serre dimension $K$4, defined as the smallest $K$5 such that $K$6 is covered by finitely many subspaces of combinatorial dimension at most $K$7. This refines both the Jacobson and Krull dimensions, with strict inequalities possible in both directions. The key properties established are: $K$8 iff $K$9 is semilocal; monotonicity under quotients and finite homomorphisms; invariance under quotient by ideals inside the Jacobson radical; and $R$0 for completions. The semilocal case is the base of the induction, and the completion formula is what drives the localization–completion step.

Elementary subgroups and perfectness

A substantial technical portion develops the theory of isotropic pinnings — data consisting of a Tits-index-type map $R$1 between absolute and relative root systems, root subgroup schemes $R$2, a centralizer $R$3, and Weyl elements $R$4 — together with their behavior under central quotients, direct products, and Weil restrictions. The elementary subgroup is defined for arbitrary commutative unital $R$5-algebras in infinitary positive categories via a generating morphism $R$6, and is shown to be independent of choices, normalized by all scheme automorphisms, and stable under products, central quotients, and Weil restriction.

A notable standalone result is that the derived subgroup $R$7 is always perfect, even when $R$8 itself is not (the latter fails only over residue fields $R$9 with absolute components of type $\mathrm{K}_1$0 or $\mathrm{K}_1$1). The proof for $\mathrm{K}_1$2 and $\mathrm{K}_1$3 is by explicit commutator calculations with root elements, exhibiting a smaller subgroup $\mathrm{K}_1$4 generated by special products of root elements that coincides with the derived subgroup. This perfectness of the derived subgroup is what upgrades the filtration result to the statement that $\mathrm{K}_1$5 is the largest perfect subgroup.

The dimension filtration

The core construction is the dimension filtration

$\mathrm{K}_1$6

where $\mathrm{K}_1$7 consists of elements elementary over every $\mathrm{K}_1$8-algebra $\mathrm{K}_1$9 with $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$0. The central commutator estimate $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$1 is proved via a localization–completion argument: an element of $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$2 becomes elementary after localizing at a single $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$3 chosen so that $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$4, while an element of $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$5 becomes elementary over the completion $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$6; a lemma modeled on Bak's method, but recast abstractly through colocalization rather than direct matrix calculation, then gives $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$7. Consequently, solvability of $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$8 reduces to solvability of $$\mathrm{K}_1(n, R, I) = \mathrm{GL}(n, R, I)/\mathrm{E}(n, R, I)$$9, which embeds into a product of $\mathrm{GL}(n, R, I)$0 for semilocal $\mathrm{GL}(n, R, I)$1. A braided crossed module argument further shows that it suffices to treat simple simply connected or adjoint group schemes, since passage along a central isogeny induces a braided crossed module on $\mathrm{GL}(n, R, I)$2-functors.

Solvability over semilocal rings

For semilocal $\mathrm{GL}(n, R, I)$3 with a globally isotropic pinning of rank at least $\mathrm{GL}(n, R, I)$4, solvability of $\mathrm{GL}(n, R, I)$5 is proved case by case over the Tits indices. Classical indices reduce to stability results for linear and unitary $\mathrm{GL}(n, R, I)$6-functors (including a new abelian-ness result for $\mathrm{GL}(n, R, I)$7 of hyperbolic-stabilized modules, proved via a Witt-cancellation and hyperbolization argument), plus a Cartan–Dieudonné-type lemma for orthogonal groups of isotropic rank $\mathrm{GL}(n, R, I)$8 over $\mathrm{GL}(n, R, I)$9-residue situations. The exceptional indices are handled by embedding suitable root subsystem group subschemes $G$0 whose indices are classical; the most involved case is $G$1, where the central product structure of $G$2 forces a two-step reduction through quaternion and quadratic form algebras, using an orthogonal-basis lemma for hermitian forms over such form algebras. The index $G$3 is cited from prior work, which is precisely why the main theorem requires $G$4 to contain a field in that case.

If $G$5 is only locally isotropic over a semilocal ring, a cardinality filtration $G$6 (elements elementary over semilocal algebras with at most $G$7 maximal ideals) yields solvability of $G$8 with solvability length depending on the number of maximal ideals. For classical groups, a uniform bound is then obtained via a decomposition lemma: for a radical power idempotent algebra $G$9 (in the pro-set category), $K$0, proved using the big cell decomposition and stability of unitary elementary groups. This is a genuine strengthening, as it removes the dependence on $K$1.

Limitations and open questions

The paper is candid about its boundaries. The full generality of $K$2 over arbitrary rings remains open, and the main theorem carries the field-containment hypothesis for that index. The relative theory is not treated: levels need not be ideals when $K$3 is not invertible and the absolute root system is non-simply-laced, and the $K$4-normal structure theorem itself is incomplete (twisted symplectic rank-$K$5 case with non-invertible $K$6; the $K$7 globally isotropic case). The dimension filtration is impredicative, quantifying over the proper class of all $K$8-algebras; the author offers a predicative replacement $K$9 valid for $2$0, at the cost of functoriality only on small algebras. A list of open problems includes: finite generation of $2$1 for finitely generated $2$2; uniform solvability bounds for locally (not globally) isotropic $2$3; the smallest commutator word annihilating locally isotropic $2$4 over semilocal rings; the rank-$2$5 isotropic case; the $2$6-normal structure theorem for isotropic twisted symplectic groups and for general locally isotropic reductive groups; and solvability of the relative locally isotropic $2$7-functor.

Conclusion

The paper transfers the Bak–Vavilov nilpotency paradigm for $2$8-functors to reductive group schemes under a purely local isotropy hypothesis, replacing Bak's explicit matrix computations with a conceptual colocalization-based localization–completion argument and supplementing it with a cardinality filtration and a uniform solvability bound for classical groups. The main structural conclusions — hypoabelian-ness of $2$9 and maximality of $\mathrm{K}_1$00 among perfect subgroups — hold in substantial generality, while solvability requires either global isotropy or a classical absolute root system, with the exceptional Tits index $\mathrm{K}_1$01 remaining the principal obstruction to a fully uniform statement over arbitrary base rings.