Koszul Algebras

Graded algebras Definition, examples, free graded algebras, tensor product and other operations on graded algebras, graded ideals, presentations of graded algebras, rewriting systems for algebras (non-commutative Gröbner bases). Graded modules Definition, examples, minimal resolution of a graded module (following Eilenberg), uniqueness of the minimal resolution. Quadratic algebras Definition, examples, the quadratic dual, definition of a Koszul algebra, examples, other characterization of Koszul algebras (including Backelin's characterization in terms of local lattices), quadratic algebras with a complete rewriting system are Koszul, operations on quadratic algebras that preserve Koszulness, some recent results.