The Diagonalizable Group Scheme D(M)

Let \(R\) be a commutative ring and \(M\) an abelian group. The diagonalizable group scheme \(D(M)\) over \(R\) admits two equivalent descriptions.

As a spectrum (construction). \[ D(M) \;=\; \operatorname{Spec} R[M], \] the spectrum of the group algebra, whose Hopf structure is group-like on the canonical basis: \(\Delta(e^m) = e^m \otimes e^m\), \(\varepsilon(e^m) = 1\), \(S(e^m) = e^{-m}\).

By its points (characterization). For any \(R\)-scheme \(T\), \[ D(M)(T) \;=\; \operatorname{Hom}_{\mathrm{grp}}\bigl(M,\, \Gamma(T, \mathcal{O}_T)^{\times}\bigr), \] the group of characters of \(M\) valued in units of the coordinate ring, with pointwise multiplication. The equivalence with the spectral description is the chain \[ \operatorname{Hom}_{R}(T, \operatorname{Spec} R[M]) \;\cong\; \operatorname{Hom}_{R\text{-alg}}(R[M], \Gamma(T)) \;\cong\; \operatorname{Hom}_{\mathrm{mon}}(M, \Gamma(T)) \;\cong\; \operatorname{Hom}_{\mathrm{grp}}(M, \Gamma(T)^{\times}), \] using affineness, the universal property of the group algebra, and the fact that a monoid homomorphism from a group lands in the units.

The points description is the one to read: it determines \(D(M)\) up to canonical isomorphism and exhibits the group law with no reference to the Hopf plumbing.

Basic examples: \(D(\mathbb{Z}) = \mathbb{G}_m\) (Multiplicative Group Scheme 𝔾_m), \(D(\mathbb{Z}/n) = \mu_n\) (Roots of Unity Group Scheme μ_n), \(D(\mathbb{Z}^n) = \mathbb{G}_m^n\) (the split torus).

A group scheme is called diagonalizable if it is isomorphic to some \(D(M)\); that property, and the anti-equivalence \(M \mapsto D(M)\), are recorded at Diagonalizable Groups and Diagonalizable Groups Are Dual to Character Groups.