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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.

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This is correct although I'd be slightly careful about the phrase 'representable functor'. It's clear that you mean 'representable in affine schemes' but, of course, sometimes the phrase 'representable functor' in general algebro-geometric setting means 'representable by a scheme'. So, of course, something like an abelian variety would not be included. This, of course, is relevant to the general setting since the functor

$$\mathsf{GrpSch}\to \cdots\hookrightarrow \mathsf{PSh}(\mathsf{AffSch})$$

(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:

  • Start here with his course notes.
  • Look then here and here for additional information from those courses (including homework)
  • Once you're done with that, you can use his notes focusing on the structure theory of reductive groups over fields
  • If you're still hungry to know about what happens beyond that case, you can read his notes on reductive groups over general schemes (which is also a good reference for the first 50 pages that records the classical theory over algebraically closed fields). This is far from contentless since a huge amount of modern number theory showing up in the Langlands program requires thinking about reductive group schemes over Dedekind domains, which already requires most of the machinery used in those notes.
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