Base change of the derived subgroup #
Let H be a commutative Hopf algebra over a field k, and let K / k be a field extension.
This file proves the canonical containment
baseChangeHopfIdeal (derivedDefiningIdeal H) ≤
derivedDefiningIdeal (baseChange H).
Since closed subgroups and their defining ideals are ordered oppositely, this says that the
derived subgroup formed after extension to K is a closed subgroup of the base change of the
derived subgroup formed over k.
The proof uses the universal property of the derived subgroup. Restriction of scalars identifies
the points of the base-changed group with the original group's points and carries the base-changed
derived subgroup to the original one. Hence every commutator lies in the base-changed subgroup;
that subgroup is normal and its pointwise quotients are commutative, so it contains the derived
subgroup formed over K.
The reverse containment, and therefore equality, requires descent for the largest Hopf ideal in the commutator kernel; it is not proved here.
Main result #
TauCeti.CommHopfAlgCat.baseChangeHopfIdeal_derivedDefiningIdeal_le: the derived subgroup formed after field extension factors through the base change of the original derived subgroup.
References #
- J. S. Milne, Algebraic Groups (2017), §6d, especially Propositions 6.17 and 6.18.
- W. C. Waterhouse, Introduction to Affine Group Schemes, Chapters 10 and 16.
This advances the base-change compatibility of G_der required in Layer 6, "Reductive and
semisimple groups", of the ReductiveGroups roadmap.
The derived subgroup formed after a field extension is contained in the base change of the original derived subgroup.
In coordinate rings, this is the displayed inclusion of defining Hopf ideals; its direction is opposite to the corresponding inclusion of closed subgroup schemes.