noncommutative geometry 2

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.

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