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 socalled categorical Galois theory, which is supposed to unify and simplify all known cases, and most recent noncommutative versions of Galoistheoretic constructions that are still to be understood categorically. Among them are HopfGalois extensions, coalgebraGalois 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: HopfGalois extensions: crossed products as cleft extensions, Schneider's theorems; differentialgeometric aspects of HopfGalois theory (strong connections, principal comodule algebras, free actions of compact quantum groups on unital C*algebras); the ChernGalois 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, predual BourbakiJacobson correspondence, "noncommutative descent", i.e., Galois condition and monadicity). Prerequisites: The course is essentially selfcontained. 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


Download  Last update  
03.12.2007 

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Part II
George Janelidze
Notes by G. Janelidze, P. Witkowski


Download  Last update  
16.08.2008 

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Part III
Tomasz Brzeziński
Notes by P. Witkowski


Download  Last update  
30.09.2008 

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