Tomorrow
I’ll start with the course Projects in non-commutative geometry
in our masterclass. The idea of this course (and its companion
Projects in non-commutative algebra run by Fred Van Oystaeyen) is
that students should make a small (original if possible) work, that may
eventually lead to a publication.
At this moment the students
have seen the following : definition and examples of quasi-free algebras
(aka formally smooth algebras, non-commutative curves), their
representation varieties, their connected component semigroup and the
Euler-form on it. Last week, Markus Reineke used all this in his mini-course
Rational points of varieties associated to quasi-free
algebras. In it, Markus gave a method to compute (at least in
principle) the number of points of the non-commutative Hilbert
scheme and the varieties of simple representations over a
finite field. Here, in principle means that Markus demands a lot of
knowledge in advance : the number of points of all connected components
of all representation schemes of the algebra as well as of its scalar
extensions over finite field extensions, together with the action of the
Galois group on them … Sadly, I do not know too many examples were all
this information is known (apart from path algebras of quivers).
Therefore, it seems like a good idea to run through Markus’
calculations in some specific examples were I think one can get all this
: free products of semi-simple algebras. The motivating examples
being the groupalgebra of the (projective) modular group
PSL(2,Z) = Z(2) * Z(3) and the free matrix-products $M(n,F_q) *
M(m,F_q)$. I will explain how one begins to compute things in these
examples and will also explain how to get the One
quiver to rule them all in these cases. It would be interesting to
compare the calculations we will find with those corresponding to the
path algebra of this one quiver.
As Markus set the good
example of writing out his notes and posting them, I will try to do the
same for my previous two sessions on quasi-free algebras over the next
couple of weeks.
Tag: geometry
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.
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.