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Let $\mathcal{G}$ be a affine algebraic group scheme(may not be reductive) over a scheme $S$. How to define a rational representaion of $\mathcal{G}$ (over $S$)? Is there always a faithful representation?

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  • $\begingroup$ What if $S$ is $\text{Spec}\ \mathbb{C}$ and $\mathcal{G}$ is an Abelian variety? Do you want to add a hypothesis that $\mathcal{G}$ is a closed subgroup scheme of $\textbf{GL}_{n,S}$, or at least that it is affine over $S?$ $\endgroup$
    – Jason Starr
    Commented Nov 12, 2017 at 21:22
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    $\begingroup$ I believe it remains an unsolved problem to determine if all smooth affine group schemes over the ring $k[\epsilon]$ of dual numbers over a field arise as a closed subgroup scheme of some ${\rm{GL}}_n$. So that device which is so useful over fields is not available more generally. Anyway, one can always contemplate $S$-group homomorphisms $\mathcal{G} \to {\rm{GL}}_n$ (even if no interesting ones exist), but calling these "rational representations" sounds weird when the base isn't a field. $\endgroup$
    – nfdc23
    Commented Nov 12, 2017 at 22:29
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    $\begingroup$ I strongly agree with nfdc23's comment and especially the last part. I'm unaware of any setting in which one has to consider "rational representations" over a base that isn't at least a commutative ring such as a field. For example, since Jantzen's book is about representation theory, he has no need to work in greater generality over arbitrary schemes. A field is usually complicated enough. $\endgroup$
    – Jim Humphreys
    Commented Nov 13, 2017 at 2:02
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    $\begingroup$ For further discussion see Aside 9.4 in Chapter 9, in Milne, "Basic Theory of Affine Group Schemes": www.jmilne.org/math/CourseNotes/AGS.pdf $\endgroup$
    – Geordie Williamson
    Commented Nov 13, 2017 at 8:49

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I'm not sure what your current sources are, but the definitions are laid out clearly in SGA3 (by Demazure and Grothendieck) and similarly in the book by Demazure and Gabriel, Groupes algebriques (North-Holland, 1970) which was later published in an English translation. (Their designation of this book as "Tome I" is of course unfortunate, since it had no sequel.)

In Demazure-Gabriel, one finds for example an explicit statement about the existence (over a field) of a faithful linear representation, in the affine case: see II, 5.2. This is far into their book but is fairly elementary, just relying on the basic notions.

For a treatment heavily influenced by Demazure-Gabriel (or SGA3), you can also consult the early sections of Jantzen's book Representations of Algebraic Groups (Academic Press, 1987; 2nd enlarged edition, Amer. Math. Soc., 2003). See especially I.2 for the notion of rational representation of a group scheme (over any commutative ring).

I should add that Jim Milne has developed a modern textbook version of all this, probably published by now; check his webpage for details: http://www.jmilne.org/math/

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