Are group schemes in Char 0 reduced? (YES) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:41:40Z http://mathoverflow.net/feeds/question/22553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22553/are-group-schemes-in-char-0-reduced-yes Are group schemes in Char 0 reduced? (YES) jlk 2010-04-26T02:27:34Z 2010-09-21T04:36:46Z <p>A Theorem of Cartier (e.g. <a href="http://books.google.com/books?id=-5weWX_YD6sC&amp;printsec=frontcover&amp;dq=curves+on+an+algebraic+surface&amp;source=bl&amp;ots=1r4O86OtJy&amp;sig=PIH5b5EipbZYJWXrWy77NvltSP4&amp;hl=en&amp;ei=4PjUS9uMKMOC8gbbi_mCDA&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=3&amp;ved=0CBIQ6AEwAg#v=onepage&amp;q&amp;f=false" rel="nofollow"> Mumford</a>, Lecture 25) states that every separated, finite type group scheme \$G/k\$ over a field \$k\$ of characteristic \$0\$ is reduced. Does this result remain valid if we drop the assumption that \$G/k\$ is separated and of finite type?</p> <p>Frans Oort (MR0206005) observed that one can use limit formalism to argue that every affine group scheme over \$k\$ is reduced.</p> <p><strong>Edit:</strong> BCnrd pointed out that group schemes over a field are automatically separated. Furthermore, the proof of Cartier's Theorem in Mumford's book remains valid for a locally Noetherian \$k\$-group scheme.</p> http://mathoverflow.net/questions/22553/are-group-schemes-in-char-0-reduced-yes/22556#22556 Answer by Harry Gindi for Are group schemes in Char 0 reduced? (YES) Harry Gindi 2010-04-26T03:04:16Z 2010-04-26T04:15:02Z <p>BCnrd posted:</p> <blockquote> <p>Every group scheme over a field is separated: rational points are closed immersions, and the diagonal is the base change of the identity section. Also, a connected group locally of finite type over field k is of finite type (use geometric connectedness and pass to the algebraic closure of k), whence smoothness follows for characteristic 0 in the locally of finite type case. (The proof of Cartier's theorem works in the locally of finite type case over a field of characteristic 0, so this reasoning is silly.) Any noetherian group scheme over field of characteristic 0 is formally smooth: the completion at 1 is a formal group of finite dimension, and Cartier's proof works in formal case (use formal Lie theory without a smoothness hypothesis!), or use Theorem 3.3ff Exp. VII of SGA3. Then translate and extend base field. QED</p> </blockquote> http://mathoverflow.net/questions/22553/are-group-schemes-in-char-0-reduced-yes/22573#22573 Answer by LRG for Are group schemes in Char 0 reduced? (YES) LRG 2010-04-26T07:59:12Z 2010-04-26T13:21:22Z <p>The answer is yes - <em>every</em> group scheme over a field of characterstic zero is reduced: see <a href="http://www.math.u-psud.fr/~biblio/numerisation/docs/P_PERRIN-109/pdf/P_PERRIN-109.pdf" rel="nofollow">Schémas en groupes quasi-compacts sur un corps et groupes henséliens</a> (especially Thm. 2.4 in part II and Thm. 1.1 and Cor. 3.9 in part V of the 1st part), and for a summary of the relevant results see 4.2 (in particular 4.2.8) of <a href="http://archive.numdam.org/article/BSMF_1976__104__323_0.djvu" rel="nofollow">Approximation des schémas en groupes, quasi compacts sur un corps</a>.</p>