Base change and adjoint semisimple affine groups

An adjoint semisimple affine group has trivial scheme-theoretic center. Formation of the center commutes with extension of the ground field, and field extensions are faithfully flat, so adjointness is unchanged by scalar extension whenever the base-changed group is supplied with its semisimple structure.

The semisimplicity hypothesis on the base-changed object is explicit. Its construction is a separate structure theorem: this file proves that no further argument about the center is needed once that hypothesis is available. The result concerns the full center scheme, not only its points over either field.

Main declaration

• TauCeti.adjointSemisimpleCommHopfAlgProperty.baseChange_iff: adjointness of a semisimple affine group is equivalent to adjointness after a field extension.

References

• J. S. Milne, Algebraic Groups (2017), §§1.k and 21.4.
• T. A. Springer, Linear Algebraic Groups, §9.6.

theorem TauCeti.adjointSemisimpleCommHopfAlgProperty.baseChange_iff {k K : Type u} [Field k] [Field K] [Algebra k K] (H : SemisimpleCommHopfAlgCat k) (hsemisimple : semisimpleCommHopfAlgProperty K H.obj.baseChange) : adjointSemisimpleCommHopfAlgProperty K { obj := H.obj.baseChange, property := hsemisimple } ↔ adjointSemisimpleCommHopfAlgProperty k H

Adjointness is preserved and reflected by a field extension.

The proof that the base-changed finite-type Hopf algebra is semisimple is kept as an explicit hypothesis, since preservation of semisimplicity is logically separate from the center calculation.