Course Materials
Section outline
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Lectures 1-2 contain an overview of the course, the definitions of Noetherian and Artin rings, and examples: Weyl algebras, group algebras and universal enveloping algebras
Lectures 3-4 cover polycyclic groups, sufficient conditions for a ring to be Noetherian, filtered and graded rings.
Lectures 5-7 cover the theory of Artinian rings: Jacobson radical, left-primitive rings and modules, and the Artin- Wedderburn theorem.
Lectures 8 and 9 cover non-commutative localisation: the Ore condition, left localisable sets, Goldie rings.
Lectures 10 and 11 cover dimension theory of modules and Krull dimension of graded modules when the graded ring is commutative.
Lectures 12-13 cover Rees rings and good filtrations of modules.
Lectures 14-15 cover the Poisson bracket of the graded ring of the Weyl algebra and a proof of Bernsteins inequality.
Lecture 16 gives the reduction of Gabber's theorem to the local case.
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This problem sheet is based on lectures in weeks 1 and 2: Group rings, Weyl algebra and Noetherian rings.
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This problem sheet is based on lectures from weeks 3 and 4, the structure theory of Artinian rings.
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This problem sheet is based on material in lectures from weeks 5 and 6: Ore rings and basic properies of dimension of modules.
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This problem sheet covers material from lectures in weeks 6,7 and 8:
Rees rings, good filtration, Poisson bracket and Bernstein's inequality.