4 Scheme theory
4.1 Correspondence between affine group schemes and Hopf algebras
We want to show that affine group schemes correspond to Hopf algebras. We must decide what this means mathematically.
We choose to interpret this as lifting Spec to a fully faithful functor from Hopf algebras to group schemes, with essential image affine group schemes.
An alternative would have been to lift the Gamma-Spec adjunction to an adjunction between Hopf algebras and affine group schemes. This is unfortunately much harder to do over an arbitrary scheme, so we leave this as future work.
4.1.1 Spec of an algebra
Spec is a contravariant functor from the category of \(R\)-algebras to the category of schemes over \(\operatorname{Spec}_R\).
Spec is a fully faithful contravariant functor from the category of \(R\)-algebras to the category of schemes over \(\operatorname{Spec}_R\), preserving all limits.
\(\operatorname{Spec}: \operatorname{Ring}\to \operatorname{Sch}\) is a fully faithful contravariant functor which preserves all limits, hence so is \(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) by Proposition 1.1.2 (alternatively, by Proposition 1.1.1).
4.1.2 Spec of a bialgebra
Spec is a contravariant functor from the category of \(R\)-bialgebras to the category of monoid schemes over \(\operatorname{Spec}_R\).
Spec is a fully faithful contravariant functor from the category of \(R\)-bialgebras to the category of monoid schemes over \(\operatorname{Spec}_R\).
\(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is a fully faithful contravariant functor preserving all limits according to Proposition 4.1.1, therefore \(\operatorname{Spec}: \operatorname{Bialg}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) too is fully faithful according to 1.2.1.
If \(A\) is a cocommutative bialgebra over \(R\), then \(\operatorname{Spec}A\) is a commutative monoid scheme.
Diagrams are the same up to identifying \(\operatorname{Spec}(A \otimes A)\) with \(\operatorname{Spec}A \otimes \operatorname{Spec}A\).
4.1.3 Spec of a Hopf algebra
Spec is a contravariant functor from the category of \(R\)-Hopf algebras to the category of group schemes over \(\operatorname{Spec}_R\).
Spec is a fully faithful contravariant functor from the category of \(R\)-Hopf algebras to the category of group schemes over \(\operatorname{Spec}_R\).
\(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is a fully faithful contravariant functor preserving all limits according to Proposition 4.1.1, therefore \(\operatorname{Spec}: \operatorname{Hopf}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) too is fully faithful according to 1.2.1.
4.1.4 Essential image of Spec on Hopf algebras
The essential image of \(\operatorname{Spec}: \operatorname{Ring}_R \to \operatorname{Sch}_{\operatorname{Spec}R}\) is precisely affine schemes over \(\operatorname{Spec}R\).
Direct consequence of Proposition 1.1.3.
The essential image of \(\operatorname{Spec}: \operatorname{Bialg}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) is precisely affine monoid schemes over \(\operatorname{Spec}R\).
The essential image of \(\operatorname{Spec}: \operatorname{Hopf}_R \to \operatorname{GrpSch}_{\operatorname{Spec}R}\) is precisely affine group schemes over \(\operatorname{Spec}R\).
4.2 Diagonalisable groups
Let \(G\) be a commutative group and \(S\) a base scheme. The diagonalisable group scheme \(D_S(G)\) is defined as the base-change of \(\operatorname{Spec}{\mathbb Z}[G]\) to \(S\). For a commutative ring \(R\), we write \(D_R(G) := D_{\operatorname{Spec}R}(G)\).
An algebraic group \(G\) over \(\operatorname{Spec}R\) is diagonalisable if it is isomorphic to \(D_R(G)\) for some commutative group \(G\).
Let \(R\) be a commutative ring and \(M\) an abelian monoid. Then \(D_R(M)\) is isomorphic to \(\operatorname{Spec}R[M]\).
Ask any toddler on the street.
An algebraic group \(G\) over a field \(k\) is diagonalizable if and only if \(\Gamma (G)\) is spanned by its group-like elements.
See Theorem 12.8 in [ 2 ] .
Let \(R\) be a domain. The functor \(D_R(G) := G \rightsquigarrow \operatorname{Spec}R[G]\) from the category of groups to the category of group schemes over \(\operatorname{Spec}R\) is fully faithful.
Let \(S\) be a scheme. Let \(M, N\) be commutative monoids and \(f : M \to N\) a monoid hom. Then the map \(D_S(f) : D_S(N) \to D_S(M)\) is affine.
\(\operatorname{Spec}f : \operatorname{Spec}{\mathbb Z}[N] \to \operatorname{Spec}{\mathbb Z}[M]\) is affine, since it’s a morphism of affine schemes. Therefore \(D_S(f)\) is affine, as affine morphisms are preserved under base change.
Let \(S\) be a scheme. Let \(M, N\) be commutative monoids and \(f : M \to N\) a surjective monoid hom. Then the map \(D_S(f) : D_S(N) \to D_S(M)\) is a closed embedding.
Since \(f\) is surjective, the corresponding map \(\hat f : {\mathbb Z}[M] \to {\mathbb Z}[N]\) is surjective too. Hence \(\operatorname{Spec}\hat f : \operatorname{Spec}{\mathbb Z}[N] \to \operatorname{Spec}{\mathbb Z}[M]\) is a closed embedding. Therefore \(D_S(f)\) is a closed embedding, as closed embeddings are preserved under base change.
Let \(S\) be a scheme. Let \(G, H\) be abelian groups and \(f : G \to H\) an injective group hom. Then the map \(D_S(f) : D_S(H) \to D_S(G)\) is faithfully flat.
Since \(f\) is injective, \({\mathbb Z}[H]\) is a free module over \({\mathbb Z}[G]\) by Proposition ??, hence the map \({\mathbb Z}[G] \to {\mathbb Z}[H]\) is faithfully flat and so is \(\operatorname{Spec}{\mathbb Z}[H] \to \operatorname{Spec}{\mathbb Z}[G]\). Therefore \(D_S(f)\) is faithfully flat, as faithfully flat morphisms are preserved under base change.
Let \(S\) be a scheme. Let \(G, H\) be abelian groups and \(f : G \to H\) an injective group hom. Then the map \(D_S(f) : D_S(H) \to D_S(G)\) is faithfully flat.
Since \(f\) is injective, \({\mathbb Z}[H]\) is a free module over \({\mathbb Z}[G]\) by Proposition 2.3.10, hence the map \({\mathbb Z}[G] \to {\mathbb Z}[H]\) is faithfully flat and so is \(\operatorname{Spec}{\mathbb Z}[H] \to \operatorname{Spec}{\mathbb Z}[G]\). Therefore \(D_S(f)\) is faithfully flat, as faithfully flat morphisms are preserved under base change.
Let \(R\) be a domain. Let \(G\) be an abelian group. If \(H\) is a closed subgroup of \(D_R(G)\), then \(H\) is a diagonalisable group scheme.
\(H\) is a closed subscheme of an affine scheme, hence it is affine. By Proposition 4.1.10, write \(H = \operatorname{Spec}A\) where \(A\) is a \(R\)-Hopf algebra. The closed subgroup embedding \(H \hookrightarrow D_R(G)\) becomes a surjective bialgebra hom \(R[G] \to A\) by Propositions 4.2.3 and 4.1.7. By Proposition 2.3.25, \(A\) is a diagonalisable bialgebra and therefore \(H\) is a diagonalisable group scheme.
Let \(G\) be an abelian group with an element of torsion \(n\). Let \(R\) be a commutative ring with \(n\) invertible. Then \(D_R(G)\) is disconnected.
Say \(x \in G\) is such that \(x^n = 1\). Then
is such that \(e ^2 = e\). We are done by Proposition 4.2.3.
4.3 The torus
The split torus \({\mathbb G}_m^n\) over a scheme \(S\) is the pullback of \(\operatorname{Spec}{\mathbb Z}[x_1^{\pm 1}, \dotsc , x_n^{\pm 1}]\) along the unique map \(S \to \operatorname{Spec}{\mathbb Z}\).
Let \(R\) be a domain. The functor \(G \rightsquigarrow \operatorname{Spec}R[G]\) from the category of groups to the category of group schemes over \(\operatorname{Spec}R\) is a group isomorphism on hom sets.
For a group scheme \(G\) over \(S\), the character lattice of \(G\) is
An element of \(X(G)\) is called a character.
For a group scheme \(G\) over \(S\), the cocharacter lattice of \(G\) is
An element of \(X^*(G)\) is called a cocharacter or one-parameter subgroup.
Let \(R\) be a domain and \(G\) be a commutative group. Then \(X(\operatorname{Spec}R[G]) = G\).
Let \(R\) be a domain and \(G\) be a commutative group. Then \(X^*(\operatorname{Spec}R[G]) = \operatorname{Hom}(G, {\mathbb Z})\).
Let \(G\) be a torus of dimension \(n\) over a domain \(R\). Then \(X(G) = {\mathbb Z}^n\).
Let \(G\) be a torus of dimension \(n\) over a domain \(R\). Then \(X^*(G) = \operatorname{Hom}({\mathbb Z}^n, {\mathbb Z})\).
Let \(R\) be a domain and \(G\) a group scheme over \(\operatorname{Spec}R\). Then there is a \({\mathbb Z}\)-valued perfect pairing between \(X(G)\) and \(X^*(G)\).
The character-cocharacter pairing on a torus is perfect.
Transfer everything across the isos \(X({\mathbb G}_m^n) = {\mathbb Z}^n, X*({\mathbb G}_m^n) = \operatorname{Hom}({\mathbb Z}^n, {\mathbb Z})\).
Let \(R\) be a domain. Let \(T\) be a split torus over \(R\). Let \(G\) be a diagonalisable group scheme over \(R\) and let \(\phi : T \to G\) be a homomorphism. Then the (scheme theoretic) image of \(\phi \) is a split torus over \(R\) and the maps
are group homomorphisms, and \(\hat{\phi }\) is fpqc. Furthermore, if \(T = D_R(H), G = D_R(I), \phi = D_R(f)\) for \(H\) a finitely generated free abelian group, \(I\) an abelian group, \(f : I \to H\) a group hom, then \(\operatorname {im} \phi \cong D_R(\operatorname {im}(f))\).
By fullness of \(D_R\) (Proposition 4.2.5), it’s enough to handle the case where \(T = D_R(H), G = D_R(I), \phi = D_R(f)\) for \(H\) a finitely generated free abelian group, \(I\) an abelian group, \(f : I \to H\) a group hom.
Then \(\operatorname {im}(f)\) is a subgroup of the finitely-generated free abelian group \(H\), hence itself a finitely-generated free abelian group (since free is equivalent to torsion-free for finitely-generated abelian groups, and a subgroup of a torsion-free group is torsion-free).
Let \(R\) be a commutative ring of characteristic zero. Let \(T\) be a split torus. If \(H \subseteq T\) is a connected closed subgroup, then \(H\) is a split torus.
By assumption, write \(T \cong D_k[G]\) for \(G\) a free abelian group. By Proposition 4.2.10, \(H\) is a diagonalisable group scheme, say \(H \cong D_k(I)\) for \(I\) an abelian group. Since \(H\) is a closed subscheme, the map \(G \to I\) is surjective, so \(I\) is a finitely-generated abelian group. Since \(H\) is connected, Proposition 4.2.11 says \(I\) is torsion-free, hence free. Thus \(H\) is a split torus.