Essential Image of Spec Is Affine Group Schemes
The essential image of the Spec functor on Hopf algebras
\[ \operatorname{hopfSpec}_R : \mathsf{Grp}((\mathsf{CommAlgCat}\,R)^{\mathrm{op}}) \longrightarrow \mathsf{Grp}(\mathsf{Over}(\operatorname{Spec} R)) \]
consists exactly of the affine group schemes over \(\operatorname{Spec} R\) — equivalently, those group objects \(G\) in \(\mathsf{Over}(\operatorname{Spec} R)\) whose underlying scheme \(G\) is affine. Combined with full faithfulness, this yields an equivalence of categories
\[ \mathsf{Grp}((\mathsf{CommAlgCat}\,R)^{\mathrm{op}}) \;\;\simeq\;\; \bigl\{\, G \in \mathsf{Grp}(\mathsf{Over}(\operatorname{Spec} R)) \,:\, G \text{ is affine} \,\bigr\}. \]
Proof sketch.
The corresponding statement at the algebra level: the essential image
of \(\operatorname{algSpec}_R\) is the affine schemes over
\(\operatorname{Spec} R\). This follows from the standard equivalence
between commutative \(R\)-algebras and affine schemes over
\(\operatorname{Spec} R\) (one direction by global sections, the other
by \(\operatorname{Spec}\)).
To lift to the Hopf level: given an affine group object \(G\) in \(\mathsf{Over}(\operatorname{Spec} R)\), the underlying scheme is \(\operatorname{algSpec}_R(A)\) for the commutative \(R\)-algebra \(A = \Gamma(G, \mathcal{O}_G)\) (the coordinate algebra). The group object structure on \(G\) endows \(A\) with comultiplication, counit, and antipode (each is the pullback of the corresponding scheme-level morphism), making \(A\) a commutative \(R\)-Hopf algebra. Under [[node:affine_group_schemes.hopf_spec_grpobj_via_yoneda|the Yoneda construction]] of \(\operatorname{hopfSpec}_R\), the \(\operatorname{GrpObj}\) it produces on \(\operatorname{algSpec}_R(A)\) matches the original \(\operatorname{GrpObj}\) on \(G\) (because both are uniquely determined by the convolution group structure on \(\operatorname{Hom}_{R\text{-Alg}}(A, -)\)).
The inverse functor at the algebra level is the global-sections functor
\(\Gamma : G \mapsto \mathcal{O}(G)\); see
the coordinate Hopf algebra.
\(\square\)