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 **l**G of finite

groups, polynomial algebras, free and tensor-algebras, path algebras

**l**Q 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.