Smoothness of the prime-field short-root G₂ carrier

The short-root type-G₂ carrier over 𝔽₃ is smooth. The coordinate algebras of its four numbered root subgroups and rank-two weight torus are reduced. Over the perfect ground field 𝔽₃, the common-kernel quotient they generate is therefore smooth. This concerns the actual prime-field carrier, so its scalar extensions are smooth as well.

References

• W. C. Waterhouse, Introduction to Affine Group Schemes, §11.4.
• J. S. Milne, Algebraic Groups (2017), Proposition 1.26 and Corollary 1.27.

instance TauCeti.G2ShortRoot.PrimeField.algebraSmooth_carrierAlgebra : Algebra.Smooth (ZMod 3) ↑carrierAlgebra

The coordinate algebra of the short-root type-G₂ carrier over 𝔽₃ is smooth.

instance TauCeti.G2ShortRoot.PrimeField.smooth_groupScheme : AlgebraicGeometry.Smooth groupScheme.X.hom

The structural morphism of the prime-field short-root type-G₂ carrier is smooth.