Galois Theory of Commutative Rings


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Classical Galois theory of fields (as met in an undergraduate course) was generalised to commutative rings around 1960 by Auslander, Goldman, Chase, Harrison and Rosenburg. About the same time, Grothendieck developed the theory of  ́etale coverings of schemes which gave a more geometric viewpoint. In the ensuing half century the general framework has been expanded to many more settings including derived algebra.

This course will discuss the basic ideas that are common to these. The lectures will explore the following themes (the depth of coverage might depend on the participant’s backgrounds and interests).

  • Review of the field theory version of Galois Theory: normal extensions, separable extensions, Normal Basis Theorem, the Galois Correspondence, number fields, Kummer Theory and abelian extensions.
  • Definition of Galois extensions of commutative rings: separable algebras, idempotents, ramification and Kähler differentials, étale extensions.
  • Properties of Galois extensions, the Galois Correspondence, homological algebra, Azumaya algebras.
  • Cohomological properties, Brauer groups, Picard groups, Harrison groups for abelian extensions.
  • Algebraic geometric interpretation: étale coverings.
  • Further generalisations: Derived algebra, topological Galois extensions, ...