Differentiated torus actions

A tangent vector coming from a split torus over a commutative ring has eigenspaces spanning every rational representation. Over an algebraically closed field this applies to every torus. The representation can have arbitrary dimension, and the homomorphism from the torus need not be a closed immersion.

The eigenspace decomposition is transported from DiagonalizableGroup.iSup_eigenspace_differential_eq_top through a splitting isomorphism. It applies in every characteristic, even when distinct characters have equal differentials.

References

• J. C. Jantzen, Representations of Algebraic Groups, I.2 and I.7.
• J. S. Milne, Algebraic Groups (2017), Chapters 12 and 22.

theorem TauCeti.splitTorusCommHopfAlgProperty.iSup_eigenspace_differential_eq_top {k : Type u} [CommRing k] {T : FiniteTypeCommHopfAlgCat k} {H : Type v} [CommRing H] [Bialgebra k H] (V : Type w) [AddCommGroup V] [Module k V] [Comodule k H V] (hT : splitTorusCommHopfAlgProperty k T) (f : H →ₐc[k] ↑T.obj) (d : Derivation k (↑T.obj) (Bialgebra.CounitAlgebra k (↑T.obj) k)) : ⨆ (μ : k), (Comodule.differential ((derivationComp f) d)).eigenspace μ = ⊤

A tangent vector coming from a split torus has eigenspaces spanning any rational representation, without a dimension or characteristic restriction.

theorem TauCeti.torusCommHopfAlgProperty.iSup_eigenspace_differential_eq_top {k : Type u} [Field k] {T : FiniteTypeCommHopfAlgCat k} {H : Type v} [CommRing H] [Bialgebra k H] (V : Type w) [AddCommGroup V] [Module k V] [Comodule k H V] [IsAlgClosed k] (hT : torusCommHopfAlgProperty k T) (f : H →ₐc[k] ↑T.obj) (d : Derivation k (↑T.obj) (Bialgebra.CounitAlgebra k (↑T.obj) k)) : ⨆ (μ : k), (Comodule.differential ((derivationComp f) d)).eigenspace μ = ⊤

Over an algebraically closed field, a tangent vector coming from any torus has eigenspaces spanning every rational representation.