Title: Syzygies and Planar Graph Resolution
-Abstract-
In Algebraic Geometry we can understand the geometry of varieties by
studying the properties of the polynomial ring S = k[x_1, ... ,x_r] and
its ideals. To effectively study these ideals we need to study also the
graded S-modules. The easiest way to describe one of these modules is by
giving the generators and the relations among them. The theory of Syzygies
and Resolutions arose from this point of view. I will define these
objects, and then I will give a combinatorial technique to compute the
Syzygies of a S-module coming from a monomial ideal in the trivariate
polynomial ring. To a monomial ideal we will associate a graph (the
Buchberger Graph) from which we can read all the information needed to
describe the S-module associated to the given ideal. Also, this graph
allows us to give an efficient algorithm to compute the syzygies, the free
resolution, and more.
Nothing beyond the basic definition of module will be needed to follow
this talk.