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Michi’s blog » archive for March, 2008

 Tour dates

  • March 6th, 2008
  • 10:57 pm

Edited to add Galway

I’ll be doing a “US tour” in March / April. For the people who might be interested – here are my whereabouts, and my speaking engagements.

I’m booked at several different seminars to do the following:

Title: On the computation of A-infinity algebras and Ext-algebras

For a ring R, the Ext algebra Ext_R^*(k,k) carries rich information about the ring and its module category. The algebra Ext_R^*(k,k) is a finitely presented k-algebra for most nice enough rings. Computation of this ring is done by constructing a projective resolution P of k and either constructing the complex Hom(P_n,k) or equivalently constructing the complex Hom(P,P). By diligent choice of computational route, the computation can be framed as essentially computing the homology of the differential graded algebra Hom(P,P).

Being the homology of a dg-algebra, Ext_R^*(k,k) has an induced A-infinity structure. This structure, has been shown by Keller and by Lu-Palmieri-Wu-Zhang, can be used to reconstruct R from
Ext_R^{\leq 2}(k,k).

 Introduction to Algebraic Geometry (3 in a series)

  • March 4th, 2008
  • 3:37 pm

I’m going to move on with the identification of geometric objects with functions from these objects down to a field soon enough, but I’d like to spend a little time nailing down the categorical language of this association. Basically, we have two functors I and V going back and forth between two categories. And the essential statement of the last post is that these two functors form an equivalence of categories.

Now, first off in this categorical language, I want to nail down exactly what the objects are. In the category \mathcal{AV}ar_k the objects are solution sets of systems of polynomial equations. And in the category \mathcal{RA}lg_k, the objects are finitely presented Noetherian reduced k-algebras.

The functor V:\mathcal{RA}lg_k\to \mathcal{AV}ar_k acts on objects by sending an algebra R to the solution set of the polynomial equations generating the ideal in a presentation of the algebra.