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Given a group scheme $X$ over $S$, where $S$ is an arbitrary locally noetherian scheme, then how does one define the Lie algebra of $X$? And how does it behave with respect to base change?

Is there any good reference for the theory of group schemes apart from Demazure/Gabriel's book about Algebraic Groups?

All of the treatings I have encountered only care about affine schemes, often over a base field. Where can I find a more general exposition?

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As far as I remember, "Jantzen: Representations of Algebraic Groups" used to work over a general base ring and specializes (to fields, algebraic closed fields, etc.) only where necessary. –  Ralph Oct 23 '11 at 11:45
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There is also SGA3. In particular, volume I, Expose II, has a section on the Lie algebra of a group scheme. –  ulrich Oct 23 '11 at 13:05
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Beyond the definition, it's also interesting to ask whether Lie algebra theory plays any essential role outside the traditional affine framework. Aside from that, a couple of added tags would be useful: reference-request and lie-algebras; maybe also algebraic-groups. –  Jim Humphreys Oct 23 '11 at 14:19

4 Answers 4

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To elaborate on a comment by ulrich: SGA3 Exp. 2, section 4 treats Lie algebras of arbitrary group-valued functors over an arbitrary scheme (no locally noetherian hypothesis). I'm not sure what results you want with respect to base change, but most will follow straightforwardly from some combination of Definition 1.1 and Proposition 3.4.

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I think you'll find quite J.S. Milne's Algebraic Groups, Lie Groups, and their Arithmetic Subgroups course notes pretty informative.

There are also a discussion of group schemes in the book I'm currently reading ; Bosch, Lütkebohmert and Raynaud's "Néron models", but it's not the main topic.

I'm pretty certain Mumford discusses group schemes in his "Abelian varieties", but I'm not sure on what base.

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For affine group schemes you can have a look to Introduction to affine group schemes by Waterhouse. The approach is by functor-of-points, and derivations and associated Lie algebra are also discussed.

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the book by sancho de salas 'grupos algebraicos y theoria de invariantes' seems to work in enough generality.

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