Cartier duality preserves rank

For a finite locally free commutative affine group scheme over a commutative ring, the Cartier dual has the same scheme-theoretic rank at every point of the base. Thus the statement also applies when rank varies between components, and does not require the base to be reduced or the rank to be invertible.

We compare the structural morphism with the spectrum of the coordinate Hopf algebra using the existing Hopf-spectrum anti-equivalence. The coordinate algebra of the Cartier dual is the finite linear dual, whose local rank equals that of the original finite projective module. This rank identity is used when passing between finite subgroup schemes and their duals.

References

• W. C. Waterhouse, Introduction to Affine Group Schemes, Chapter 2.
• M. Demazure and A. Grothendieck, Schémas en groupes (SGA 3), Exposé VIIA, §3.3.

@[simp]

theorem TauCeti.FiniteLocallyFreeCommAffineGroupSchemeCat.finrank_eq_rankAtStalk_coordinateHopfAlgebra (R : Type u) [CommRing R] (G : FiniteLocallyFreeCommAffineGroupSchemeCat ↧R) (x : PrimeSpectrum R) : AlgebraicGeometry.Scheme.Hom.finrank G.obj.obj.X.hom x = Module.rankAtStalk (↑(coordinateHopfAlgebra R G).obj) x

The scheme-theoretic rank of a finite locally free commutative affine group scheme is the local rank of its coordinate Hopf algebra.

@[simp]

theorem TauCeti.FiniteLocallyFreeCommAffineGroupSchemeCat.finrank_cartierDual (R : Type u) [CommRing R] (G : FiniteLocallyFreeCommAffineGroupSchemeCat ↧R) : AlgebraicGeometry.Scheme.Hom.finrank (cartierDual R G).obj.obj.X.hom = AlgebraicGeometry.Scheme.Hom.finrank G.obj.obj.X.hom

Cartier duality preserves the entire locally constant rank function over the affine base.