Finite generation of elementary subgroups

Establish that the elementary subgroup \(\Elem_G(K)\) is finitely generated whenever the ring \(K\) is finitely generated.

Background

The paper defines elementary subgroups $\Elem_G(K)$ for reductive group schemes with local isotropic rank at least two and proves structural results concerning their perfectness, normality, and associated $\KFunc_1$-functors. It leaves unresolved whether finite generation of the base ring implies finite generation of the elementary subgroup.

References

Finally, let us list several open problems. Show that \Elem_G(K) is finitely generated if K is finitely generated.

— Nilpotency of locally isotropic $ \mathrm{K}_1 $-functor  (2608.17488 - Voronetsky, 18 Aug 2026) in Section 10, Main result and concluding remarks, final list of open problems