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Comatrix corings applied to weak and partial Galois theory

Friday, 23 September, 2005 - 17:00
Campus: Brussels Humanities, Sciences & Engineering campus
Faculty: Science and Bio-engineering Sciences
Erwin De Groot
phd defence

Corings can be viewed as coalgebras over noncommutative
rings, in fact they are coalgebras in the monoidal category of bimodules
over a (non-commutative) ring. A beautiful and important application of
corings is an elegant reformulation of descent and Galois theory. We have
developed descent and Galois theory over comatrix corings, that
generalize the Sweedler canonical coring. We apply this in two particular
situations: we first discuss Galois theory of weak Hopf algebras,
generalizing Chase and Sweedler's Galois theory for classical Hopf
algebras. A tool that is introduced first is the notion of weak entwining
structures. The second application concerns partial actions by groups; in
the case where the actions are defined everywhere, we recover the
classical Galois theory over commutative rings. We have also discussed a
class of infinite comatrix corings, that can be constructed as colimits of
finite comatrix corings. In the final chapter, we introduce Frobenius
functors of the second kind, and give some applications.