noncommutative geometry

Today I did prepare my lectures for tomorrow for the NOG master-class on non-commutative geometry. I\’m still doubting whether it is worth TeXing my handwritten notes. Anyway, here is what I will cover tomorrow :

- Examples of l-algebras (btw. l is an arbitrary field) : matrix-algebras, group-algebras lG of finite groups, polynomial algebras, free and tensor-algebras, path algebras lQ of a finite quiver, coordinaterings O(C) of affine smooth curves C etc.
- Refresher on homological algebra : free and projective modules, exact sequences and complexes, Hom and Ext groups and how to calculate them from projective resolutions, interpretation of Ext^1 via (non-split) short exact sequences and stuff like that.
- Hochschild cohomology and noncommutative differential forms. Bimodules and their Hochschild cohomology, standard complex and connection with differential forms, universal bimodule of derivations etc.
- Non-commutative manifolds. Interpretation of low degree Hochschild cohomology, characterization of non-commutative points as separable l-algebras and examples. Formally smooth algebras (non-commutative curves) characterised by the lifting property for square-free extensions and a proof that formally smooth algebras are hereditary.

Next week I will cover the representation varieties of formally smooth algebras and the semigroup on their connected (or irreducible) components.

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