Some structure theorems for algebraic groups.

*(English)*Zbl 1401.14195
Can, Mahir Bilen (ed.), Algebraic groups: structure and actions. 2015 Clifford lectures on algebraic groups: structures and actions, Tulane University, New Orleans, LA, USA, March 2–5, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2601-9/hbk; 978-1-4704-3751-0/ebook). Proceedings of Symposia in Pure Mathematics 94, 53-126 (2017).

These are extended notes of a course given at Tulane University for the 2015 Clifford Lectures about structure results for group schemes of finite type over a field.

A first structure result asserts that every algebraic group \(G\) (over a field \(k\)) has a largest connected normal subgroup scheme \(G^0\) and the quotient \(G/G^0\) is finite and étale. The main goal of this text is to present the following two more advanced results: Theorem 1. Every \(G\) has a smallest normal subgroup scheme \(H\) such that \(G/H\) is affine. Moreover, \(H\) is smooth, connected and contained in the center of \(G^0\); in particular \(H\) is commutative. Also, \(H\) is the largest subgroup scheme such that \(\mathcal{O}(H)=k\). Theorem 2. Every \(G\) has a smallest normal subgroup scheme \(N\) such that \(G/N\) is proper. Moreover, \(N\) is affine and connected. If \(k\) is perfect and \(G\) smooth, then \(N\) is smooth as well. Moreover, the formation of \(G^0\), \(H\) and \(N\) commutes with field extension. The two previous results have the following important consequences. Every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth affine (or equivalently linear) connected algebraic group. Also, every smooth connected algebraic group over a field is an extension of a linear algebraic group by an anti-affine algebraic group \(H\); i.e., every global function on \(H\) is constant. Every abelian variety is anti-affine; but the converse is false. Still, in the text it is explained how to reduce the structure of an anti-affine group over an arbitrary field to that of an abelian variety. Finally, in the text some results are presented about Picard schemes and automorphism group schemes of proper schemes. The text presents scheme-theoretic proofs of the two main results. The prerequisites are familiarity with the contents of chapters 2 to 5 of [Q. Liu, Algebraic geometry and arithmetic curves. Oxford: Oxford University Press (2002; Zbl 0996.14005)] and familiarity with some basic results on abelian varieties from [D. Mumford, Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Corrected reprint of the 2nd ed. 1974. New Delhi: Hindustan Book Agency/distrib. by American Mathematical Society (AMS); Bombay: Tata Institute of Fundamental Research (2008; Zbl 1177.14001)]. The author does not make explicit use of sheaves for the fpqc or fppf topology.

For the entire collection see [Zbl 1364.12001].

A first structure result asserts that every algebraic group \(G\) (over a field \(k\)) has a largest connected normal subgroup scheme \(G^0\) and the quotient \(G/G^0\) is finite and étale. The main goal of this text is to present the following two more advanced results: Theorem 1. Every \(G\) has a smallest normal subgroup scheme \(H\) such that \(G/H\) is affine. Moreover, \(H\) is smooth, connected and contained in the center of \(G^0\); in particular \(H\) is commutative. Also, \(H\) is the largest subgroup scheme such that \(\mathcal{O}(H)=k\). Theorem 2. Every \(G\) has a smallest normal subgroup scheme \(N\) such that \(G/N\) is proper. Moreover, \(N\) is affine and connected. If \(k\) is perfect and \(G\) smooth, then \(N\) is smooth as well. Moreover, the formation of \(G^0\), \(H\) and \(N\) commutes with field extension. The two previous results have the following important consequences. Every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth affine (or equivalently linear) connected algebraic group. Also, every smooth connected algebraic group over a field is an extension of a linear algebraic group by an anti-affine algebraic group \(H\); i.e., every global function on \(H\) is constant. Every abelian variety is anti-affine; but the converse is false. Still, in the text it is explained how to reduce the structure of an anti-affine group over an arbitrary field to that of an abelian variety. Finally, in the text some results are presented about Picard schemes and automorphism group schemes of proper schemes. The text presents scheme-theoretic proofs of the two main results. The prerequisites are familiarity with the contents of chapters 2 to 5 of [Q. Liu, Algebraic geometry and arithmetic curves. Oxford: Oxford University Press (2002; Zbl 0996.14005)] and familiarity with some basic results on abelian varieties from [D. Mumford, Abelian varieties. With appendices by C. P. Ramanujam and Yuri Manin. Corrected reprint of the 2nd ed. 1974. New Delhi: Hindustan Book Agency/distrib. by American Mathematical Society (AMS); Bombay: Tata Institute of Fundamental Research (2008; Zbl 1177.14001)]. The author does not make explicit use of sheaves for the fpqc or fppf topology.

For the entire collection see [Zbl 1364.12001].

Reviewer: Alessandro Ruzzi (Montreuil) (MR3645068)

##### MSC:

14L15 | Group schemes |

14L30 | Group actions on varieties or schemes (quotients) |

14M17 | Homogeneous spaces and generalizations |

14K05 | Algebraic theory of abelian varieties |

14K30 | Picard schemes, higher Jacobians |

14M27 | Compactifications; symmetric and spherical varieties |

20G15 | Linear algebraic groups over arbitrary fields |