2007/08, 7th semester

Course Materials
Galois structures

Classical Galois theory, whose original purpose was to give a reasonable necessary and sufficient condition for the solvability in radicals of an algebraic equation, has been modified and widely generalised many times. The list of important examples of Galois/covering theories includes: theory of covering spaces, which goes back to H. Poincaré and is often considered as the main origin of algebraic topology; étale Galois theory of schemes, due to A. Grothendieck, which plays an important role in algebraic geometry and has important applications in algebraic number theory; many others, involving several areas of abstract algebra and geometry/topology.

This course will describe the current stage of the generalisation process, including the so-called categorical Galois theory, which is supposed to unify and simplify all known cases, and most recent noncommutative versions of Galois-theoretic constructions that are still to be understood categorically. Among them are Hopf-Galois extensions, coalgebra-Galois extensions, Galois corings and comodules, and principal actions of quantum groups on C*-algebras (noncommutative principal bundles). The course is divided into three parts:

Part 1: The first goal is to give an overview of structures appearing in geometrically significant generalisations of classical Galois theory. We will focus on the following topics: Galois field extensions; Galois theory as equivalence of categories; fundamental group and universal covering; torsors and Galois cohomology; Galois context and principal fibrations; Galois context in monoidal categories; Tannakian categories.

Part 2: This part will be devoted to Galois theory in general categories and some of its examples, with necessary preliminary material from category theory included: universal properties, Yoneda embedding, and discrete fibrations; adjoint functors, monads, and algebras over monads; internal category theory and Grothendieck descent; Galois theory in general categories; Galois theory of abstract families in algebra and geometry; other examples of Galois theories.

Part 3: The final part will be devoted to noncommutative Galois theory, specifically to the following recent developments: Hopf-Galois extensions: crossed products as cleft extensions, Schneider's theorems; differential-geometric aspects of Hopf-Galois theory (strong connections, principal comodule algebras, free actions of compact quantum groups on unital C*-algebras); the Chern-Galois character (entwining structures, principal extensions, associated modules, index pairings); corings and comodules (the Sweedler coring and the coring associated to an entwining structure, comonads and associated adjoint functors, separable and split algebra extensions, corings with a grouplike element and their relation to differential graded algebras and connections); Galois comodules (corings associated to finitely generated projective modules and beyond, comatrix contexts, Galois corings, pre-dual Bourbaki-Jacobson correspondence, "noncommutative descent", i.e., Galois condition and monadicity).

Prerequisites: The course is essentially self-contained. Only basic experience and familiarity with simplest properties of rings, modules, groups, Hopf algebras, topological spaces, fibrations, and categories is expected. No particular knowledge about Galois theory is assumed.

Tomasz Maszczyk
George Janelidze
Tomasz Brzeziński
Part I
Tomasz Maszczyk
Notes by P. Witkowski
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Table of contents
  1. Galois theory
    1. Fields
    2. Morphisms of fields
    3. Polynomials
    4. Automorphisms of fields
    5. Extending isomorphisms
    6. The fundamental theorem of Galois thory
    7. The normal basis theorem
    8. Hilbert's 90 theorem
  2. Hopf-Galois extensions
    1. Canonical map
    2. Coring structure
    3. Hopf-Galois field extensions
    4. Torsors
    5. Crossed homomorphisms and G-torsors
    6. Descent theory
Part II
George Janelidze
Notes by G. Janelidze, P. Witkowski
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Table of contents
  1. Category theory
    1. How do categories appear in modern mathematics?
    2. Isomorphism and equivalence of categories
    3. Yoneda lemma and Yoneda embedding
    4. Representable functors and discrete fibrations
    5. Adjoint functors
    6. Monoidal categories
    7. Monads and algebras
    8. More on adjoint functors and category equivalences
    9. Remarks on coequalizers
    10. Monadicity
    11. Internal precategory actions
    12. Descent via monadicity and internal actions
    13. Galois structures and admissibility
    14. Monadic extensions and coverings
    15. Categories of abstract families
    16. Coverings in classical Galois theory
    17. Central extensions of groups
    18. The fundamental theorem of Galois theory
    19. Back to the classical cases
  2. Remarks on functors and natural transformations
    1. Natural transformations
    2. Limits and colimits
    3. Galois connections
Part III
Tomasz Brzeziński
Notes by P. Witkowski
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Table of contents
  1. Comonads and Galois comodules of corings
    1. Comonads
    2. Comonadic triangles and the descent theory
    3. Comonads on a category of modules. Corings
    4. Galois comodules for corings
    5. The Maszczyk Galois condition
  2. Hopf-Galois extensions of non-commutative algebras
    1. Coalgebras and Sweedler's notation
    2. Bialgebras and comodule algebras
    3. Hopf-Galois extensions and Hopf algebras
    4. Cleft extensions
    5. Hopf-Galois extensions as Galois comodules
  3. Connections in Hopf-Galois extensions
    1. Connections
    2. Connection forms
    3. Strong connections
    4. The existence of strong connections. Principal comodule algebras
  4. Principal extensions and the Chern-Galois character
    1. Coalgebra-Galois extensions
    2. Principal extensions
    3. Cyclic homology of an algebra and the Chern character
    4. The Chern-Galois character
    5. Example: the classical Hopf bration
Designed by: Pawel Witkowski