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I am trying to clarify the definition of affine group schemes. It can be seen as a representable functor from the category of algebras to the category of groups.

My question

Let $k$ be an algebraically closed field, $R := k[(x_{ij})_{i,j=1}^n,y]/(\det(x_{ij})y-1).$ Where $x_{ij}\in A$ is a commutative algebra.

Let $G(-) := Hom_{k-alg}(R,-)$ be the functor. For all $A$ is a k-algebra, $G(A)$ in the category of groups, so it should be a group, but why is that?

I want to define a multiplication in $G(A) = Hom(R,A)$, and verify that this multiplication satisfies the axiom of a group :associative law, unit and inverse.

This seems to require knowledge of Hopf algebra, but I am not sure about this part of the content. I hope to have a direct construction of this multiplication.

Please give me some tips, thank you!

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  • $\begingroup$ Try to establish a natural isomorphism $G(T)\cong \mathrm{GL}_n(T)$ for all $k$-algebras $T$. $\endgroup$
    – Lukas Heger
    Commented Mar 7 at 9:30

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