Homomorphism of Affine Group Schemes
Let \(G, H\) be affine group schemes over a commutative ring \(R\). A homomorphism \(f : G \to H\) is a morphism of \(R\)-schemes compatible with the multiplication morphisms: \[ f \circ m_G \;=\; m_H \circ (f \times f), \] i.e. a morphism of group objects in the category of \(R\)-schemes.
Unit and inversion come for free. Compatibility with the unit and inversion morphisms need not be imposed: for every \(R\)-algebra \(S\), the map \(f(S) : G(S) \to H(S)\) is a map of groups commuting with multiplication, and a multiplication-preserving map of groups automatically preserves the identity and inverses; by the functor-of-points characterization this holds at the scheme level.
Coordinate-ring translation (the bridge). For \(G = \operatorname{Spec} A\) and \(H = \operatorname{Spec} B\) with their coordinate Hopf algebras, homomorphisms \(G \to H\) correspond contravariantly to \(R\)-bialgebra homomorphisms \(B \to A\): compatibility with multiplication translates to compatibility with comultiplication, the counit condition is automatic (it is the unit compatibility), and compatibility with the antipode is automatic for bialgebra maps between Hopf algebras. This bridge decomposes as the composite of two identifications, each recorded separately:
- group-object morphisms correspond to group-object morphisms in the opposite category of algebras (Spec Functor on Hopf Algebras Is Fully Faithful);
- group-object morphisms in the opposite category of algebras correspond to bialgebra homomorphisms of the coordinate Hopf algebras — the morphism half of the dictionary between coordinate Hopf algebras and group objects.
Every use of "characters are Hopf-algebra maps" in this knowledge base passes through this bridge; see Characters Are Group-Like Elements and Convolution Is Pointwise Multiplication.