Achilleas Sinefakopoulos

August 2007 Ph.D.

First Position

Visiting assistant professor at Cornell University

Dissertation

On Some Classes of Borel Fixed Ideals and their Cellular Resolutions

Advisor

Research Area

Commutative algebra and algebraic geometry

Abstract

Let k be a field. This thesis considers some classes of Borel fixed ideals over the polynomial ring S = k[x1x2, ... , xn] and their cellular resolutions.

In Chapter 3 we work in characteristic zero and we construct a polytopal cell complex that supports a minimal graded free resolution of a Borel principal ideal in S; this includes the case of powers of the homogeneous maximal ideal (x1x2, ... , xn) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in one degree, there exists a polytopal cell complex that supports a minimal graded free resolution of I.

In Chapter 4 we work in prime characteristic p, we define the reduced horseshoe resolution and the notion of "conjoined" pairs of ideals in order to study the minimal graded free resolution of a class of p-Borel ideals. We prove that the graded Betti numbers of these ideals do not depend on the characteristic of the base field k. Also, we recover Pardue's regularity formula for this class of ideals.

Finally, in Chapter 5 we consider the lcm-lattice of strongly stable square free ideals and Borel fixed ideals generated in the same degree. We prove that this lattice is ranked and that in the case of strongly stable square free ideals it is also chain lexicographically shellable.