Tori and Galois lattices over perfect fields #
The geometric character-lattice functor is an equivalence from coordinate Hopf algebras of tori over a perfect field to continuous integral Galois lattices. On the corresponding group schemes this is the usual anti-equivalence. Fullness follows by descending equivariant geometric character maps; faithfulness and essential surjectivity hold over arbitrary fields.
The perfectness assumption is used only for descent from the algebraic closure. No choice of splitting field is needed to recover a morphism from its character map.
References #
- J. S. Milne, Algebraic Groups (2017), Theorem 12.23 and Corollary 12.24.
Over a perfect field, every equivariant morphism of character lattices is induced by a morphism of torus coordinate Hopf algebras.
The character-lattice functor classifies tori over a perfect field. Its variance here is covariant because the source category consists of coordinate Hopf algebras.