# Lie Algebra of Group Scheme

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?

• 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
• 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
• 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