I am struggling to find a reference to understand the following fact.
Let $0 \to A \to B \to C \to 0$ be a short exact sequence of abelian groups. I first apply the functor $\mathbb{C}[-]$ taking abelian groups into Hopf algebras (where co-multiplication $\mathbb{C}[M] \to \mathbb{C}[M]\otimes \mathbb{C}[M]$ is given by $m \mapsto m \otimes m$) and then Spec taking Hopf algebras into commutative affine group schemes. I would like to conclude that I get a short exact sequence $$ 0 \to \text{Spec}\,\mathbb{C}[C] \to \text{Spec}\,\mathbb{C}[B] \to \text{Spec}\,\mathbb{C}[A] \to 0. $$
I am really not familiar with this subject, thus any help to understand the problem is appreciated:
- Is the Spec functor between co-commutative Hopf algebras and commutative affine group scheme exact? Is it an equivalence?
- Is the functor $\mathbb{C}[-]$ exact? Maybe restricting the target category to that of co-commutative Hopf algebras, to get an abelian one? Am I allowed to do that?