Lie Algebra of Group Scheme - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:48:11Z http://mathoverflow.net/feeds/question/78886 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78886/lie-algebra-of-group-scheme Lie Algebra of Group Scheme Veen 2011-10-23T11:28:59Z 2011-10-24T00:37:52Z <p>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?</p> <p>Is there any good reference for the theory of group schemes apart from Demazure/Gabriel's book about Algebraic Groups?</p> <p>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?</p> http://mathoverflow.net/questions/78886/lie-algebra-of-group-scheme/78894#78894 Answer by Julien Puydt for Lie Algebra of Group Scheme Julien Puydt 2011-10-23T14:05:28Z 2011-10-23T14:05:28Z <p>I think you'll find quite J.S. Milne's <a href="http://www.jmilne.org/math/CourseNotes/ala.html" rel="nofollow">Algebraic Groups, Lie Groups, and their Arithmetic Subgroups</a> course notes pretty informative.</p> <p>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.</p> <p>I'm pretty certain Mumford discusses group schemes in his "Abelian varieties", but I'm not sure on what base.</p> http://mathoverflow.net/questions/78886/lie-algebra-of-group-scheme/78907#78907 Answer by Qfwfq for Lie Algebra of Group Scheme Qfwfq 2011-10-23T16:49:53Z 2011-10-23T16:49:53Z <p>For affine group schemes you can have a look to <a href="http://books.google.com/books?id=l0DgAIx_djoC&amp;printsec=frontcover&amp;hl=it#v=onepage&amp;q&amp;f=false" rel="nofollow"><em>Introduction to affine group schemes</em></a> by Waterhouse. The approach is by functor-of-points, and derivations and associated Lie algebra are also discussed.</p> http://mathoverflow.net/questions/78886/lie-algebra-of-group-scheme/78933#78933 Answer by Yosemite Sam for Lie Algebra of Group Scheme Yosemite Sam 2011-10-23T23:10:05Z 2011-10-23T23:10:05Z <p>the book by sancho de salas 'grupos algebraicos y theoria de invariantes' seems to work in enough generality.</p> http://mathoverflow.net/questions/78886/lie-algebra-of-group-scheme/78937#78937 Answer by S. Carnahan for Lie Algebra of Group Scheme S. Carnahan 2011-10-24T00:37:52Z 2011-10-24T00:37:52Z <p>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.</p>