Again I

spend the whole morning preparing my talks for tomorrow in the master

class. Here is an outline of what I will cover :

– examples of

noncommutative points and curves. Grothendieck’s characterization of

commutative regular algebras by the lifting property and a proof that

this lifting property in the category **alg** of all l-algebras is

equivalent to being a noncommutative curve (using the construction of a

generic square-zero extension).

– definition of the affine

scheme **rep(n,A)** of all n-dimensional representations (as always,

**l** is still arbitrary) and a proof that these schemes are smooth

using the universal property of **k(rep(n,A))** (via generic

matrices).

– whereas **rep(n,A)** is smooth it is in general

a disjoint union of its irreducible components and one can use the

sum-map to define a semigroup structure on these components when

**l** is algebraically closed. I’ll give some examples of this

semigroup and outline how the construction can be extended over

arbitrary basefields (via a cocommutative coalgebra).

–

definition of the Euler-form on **rep A**, all finite dimensional

representations. Outline of the main steps involved in showing that the

Euler-form defines a bilinear form on the connected component semigroup

when **l** is algebraically closed (using Jordan-Holder sequences and

upper-semicontinuity results).

After tomorrow’s

lectures I hope you are prepared for the mini-course by Markus Reineke on non-commutative Hilbert schemes

next week.