Section outline

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    General

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    Algebraic groups

    Schedule:  From Friday 30/01 lectures will take place every Friday 10h15-12h15 in Cramér room (Albano House 1, Floor 3), except during red days (April 3rd and May 1rst) until May the 22nd. There is another exception: on February 6th the lecture will take place from 10h00 to 12h00.

    There will be 14 up to 16 lectures of 2 hours: the course is self-contained with the 14 first lectures, lectures 15 and 16 will revolve around proof of theorems stated in lecture 14 and will propose some openings that will probably be of interest for the most algebraic part of the audience.

    If you are interested in following the lectures but cannot attend in person, there is a possibility to attend via Zoom (secret code: grpscheme) although we cannot guarantee the quality of the image or sound: the lectures are primarily intended to be attended in person, so technical issues will be resolved if the lecturer's time and skills allow. 

    Teacher: Marion Jeannin (marion.jeannin@math.su.se)

    Examination: Four problem sheets will be distributed. To pass, you need to obtain 1/4 of the score on each problem sheet, and 1/2 of the total score.

    Homework 1 (due on week 8, Friday 20/02)

    Homework 2(due on week 12, Friday 20/03)

    Homework 3 (due on week 17, Friday 24/04)

    Homework 4 (due on week 20, Friday 15/05)

    Prerequisites: Abstract algebra and topology.  A good knowledge in commutative algebra, classical algebraic geometry and category theory would be helpful but not compulsory.

    Contents: The aim of the course is to develop the theory of affine group schemes over rings from the basics and conclude with the structure theory of reductive group schemes.

    Lecture 1 Introduction: from affine algebraic sets to affine schemes seen as functor of points. Notes of lecture 1
    Lecture 2 Further constructions, algebraic and geometric properties. Notes of lecture 2 . Video of lecture 2  
    Lecture 3 Affine group schemes and their actions. Notes of lecture 3. Video of lecture 3
    Lecture 4 Representations of affine group schemes. The Lie algebra of an affine group scheme. Notes of lecture 4 Videos of lecture 4: 1/2 and 2/2    
    Lecture 5 The Lie algebra of an affine group scheme. Flatness and smoothness.Video of Lecture 5 (only the second part, I forgot to register the beginning, I apologize for the inconvenience). Notes of lecture 5
    Lecture 6 Reductive groups over algebraically closed fields I. Notes of lecture 6, Video of Lecture 6 
    Lecture 7 Reductive groups over algebraically closed fields II Notes of lecture 7    Video of lecture 7 
    Lecture 8 Reductive groups over algebraically closed fields III Notes of lecture 8     Video of lecture 8
    Lecture 9 Grothendieck topologies Notes of lecture 9 Video of Lecture 9
    Lecture 10 Torsors and descent Notes of lecture 10   Video of Lecture 10
    Lecture 11 Torsors. Generalities on reductive group schemes
    Lecture 12 Roots and coroots, a dynamic approach
    Lecture 13 Split reductive groups, pinnings 
    Lecture 14 The isogeny theorem
    Lecture 15 The existence theorem
    Lecture 16 Non-split groups

    Literature: 

    In addition of the references listed below this course has largely been built from notes of Matthieu Romagny and Philippe Gille (both in French).

    Category Theory: 

    • S. MacLane, Categories for the working mathematician,
    • J. J. Rotman, An introduction to homological algebra,
    • T. Leinster, Basic category theory.

    Commutative algebra:

    • D. Eisenbud, Commutative algebra with a view toward algebraic geometry,

    Algebraic geometry:

    • D. Eisenbud, J. Harris, The geometry of schemes,
    • U. Görtz, T. Wedhorn, Algebraic Geometry 1, Schemes with examples and exercises.

    Affine group schemes:

    • W. C. Waterhouse, Introduction to Affine Group Schemes,
    • M. Demazure and P. Gabriel, Groupes algébriques  (hard to find and in French). Half of the original book has been translated, and is available via Mathscinet at least for Stockholm University members: 
      Demazure, Michel; Gabriel, Peter, Introduction to algebraic geometry and algebraic groups. Translated from the French by J. Bell. North-Holland Publishing Co., Amsterdam-New York, 1980. xiv+357 pp.
    • SGA3 (in French), reedited.

    Reductive group schemes:

    Grothendieck topologies: