# Equivalence definition of affine (group) schemes

I am currently studying affine group schemes via Waterhouse. Since Waterhouse does not use schematic language in the first few chapters, I tried to "translate" the definitions in different languages. I just wanted to make sure that I understood correctly and did not make fundamental mistakes.

Equivalence Definition of Group Schemes

Equivalence Definition of Schemes

Restricting ourselves to just the affine case,

$$(k\text{-}alg)^{op} ~\cong~ Affsch/S ~\cong~ Rep\text{-}Func: k\text{-}alg \to Set$$ where $S = Spec(k)$. The first is the opposite category of $k$-algebras, the second is the category of affine schemes over $S$, and the third is the category of representable functors from $k$-algebra to Sets.

The $k$-algebra maps, morphism of locally ringed spaces, and natural transformations are the corresponding morphisms.

And in the case of group schemes (My definition of a group scheme is a scheme with the triple morphisms of schemes that correspond to the group axiom),

$$(Hopf~k\text{-}alg)^{op} ~\cong~ AffGrpSch/S ~\cong~Rep\text{-}Func: k\text{-}alg \to Grp$$

where $S=Spec(k)$. The first is the opposite category of Hopf $k$-algebras, the second is the category of affine group schemes over $S$, and the third is the category of representable functors from $k$-algebras to Sets.

The Hopf $k$-algebra maps, morphisms of locally ringed spaces, and natural transformations are the corresponding morphisms.

$$\mathsf{GrpSch}\to \cdots\hookrightarrow \mathsf{PSh}(\mathsf{AffSch})$$
(where I put an ellipsis since the image lies in the subcategory of sheaves on $$\mathsf{AffSch}$$ for any subcanonical topology) sending $$G$$ to its presheaf of sections on affine schemes is fully faithful, even though $$\mathsf{GrpSch}$$ contains non-affine group schemes.
Also, just an idle comment. If you are still learning about group schemes I would highly suggest you read something other than Waterhouse. I particularly recommend Milne's new book (he has some weird hangup where he wants to use $$\mathrm{MaxSpec}$$ and not $$\mathrm{Spec}$$ but just ignore that--everything there is modernly scheme theoretic) or see any of Brian Conrad's notes: