Base change of special-linear root subgroups

The elementary root maps xᵢⱼ : 𝔾ₐ → SLₙ are compatible with arbitrary base extension. The canonical identifications of the scalar-extended coordinate Hopf algebras with those constructed over the new ring preserve the normalized root maps. Thus extending an integral elementary root subgroup gives the chosen elementary root subgroup over the new base, including in positive characteristic and over rings with nilpotents.

The compatibility follows from the general-linear root-map calculation and coordinateMap_comp_rootSubgroupCoordinateMap, using the surjective determinant-one quotient. No new presentation of the root maps is introduced.

References

• B. Conrad, Reductive Group Schemes (2014), §5.1.
• J. S. Milne, Algebraic Groups (2017), §21, Example 21.2.

theorem TauCeti.SpecialLinear.baseChangeMap_rootSubgroupCoordinateMap_comp_baseChangeIso_hom (R : Type u) (K : Type (max u v)) [CommRing R] [CommRing K] [Algebra R K] {n : ℕ} {i j : Fin n} (hij : i ≠ j) : CategoryTheory.CategoryStruct.comp (CommHopfAlgCat.baseChangeMap (rootSubgroupCoordinateMap hij)) (AdditiveGroup.coordinateHopfAlgebraBaseChangeIso R K).hom = CategoryTheory.CategoryStruct.comp (coordinateHopfAlgebraBaseChangeIso R K n).hom (rootSubgroupCoordinateMap hij)

The normalized special-linear root map commutes with scalar extension under the canonical coordinate-algebra identifications. In particular, this identifies the base change of each integral root subgroup with the root subgroup over K.

theorem TauCeti.SpecialLinear.baseChangeMap_rootSubgroupCoordinateMap_comp_baseChangeIso_hom_assoc (R : Type u) (K : Type (max u v)) [CommRing R] [CommRing K] [Algebra R K] {n : ℕ} {i j : Fin n} (hij : i ≠ j) {Z : CommHopfAlgCat K} (h : AdditiveGroup.coordinateHopfAlgebra K ⟶ Z) : CategoryTheory.CategoryStruct.comp (CommHopfAlgCat.baseChangeMap (rootSubgroupCoordinateMap hij)) (CategoryTheory.CategoryStruct.comp (AdditiveGroup.coordinateHopfAlgebraBaseChangeIso R K).hom h) = CategoryTheory.CategoryStruct.comp (coordinateHopfAlgebraBaseChangeIso R K n).hom (CategoryTheory.CategoryStruct.comp (rootSubgroupCoordinateMap hij) h)

The normalized special-linear root map commutes with scalar extension under the canonical coordinate-algebra identifications. In particular, this identifies the base change of each integral root subgroup with the root subgroup over K.