Cores of Monomial Ideals

In this dissertation, we describe the cores of several classes of monomial ideals. We also find bounds on the reduction numbers of these ideals. The first class of ideals which we consider is one coming from graph theory, the strongly stable ideals of degree two. We prove that a strongly stable ideal I of degree two in k[x_1,...,x_d] has the G_d property if and only if x_{g-1}x_d is in I, where g is the height of I. We also prove that any strongly stable ideal of degree two in k[x_1,...,x_d] which has the G_d property satisfies the Artin-Nagata property AN_{d-1}. We then show that the core of such an ideal I is given by core(I)=I m^{g-1}, where g is the height of I, and m is the homogeneous maximal ideal (x_1,...,x_d). We also show that the reduction number r(I) is at most g-1. Specifically, we prove that an ideal J=J_I whose elements correspond to diagonals in the tableau associated to I is a minimal reduction of I, and that r_J(I) < g.

The other classes of ideals which we consider are zero-dimensional monomial ideals in the polynomial ring R=k[x_1,...,x_d]. We show that, if such an ideal I is an almost complete intersection, then the core of I satisfies a (d+1)-fold symmetry property coming from the generators of I, and hence that the shape of core(I) is closely related to the shape of I. Using this symmetry property, we completely describe the shape of core(I) in the case where I has a monomial minimal reduction. Then, in the two-dimensional case, we give an algorithm for computing the core of I which allows us to prove that the minimal number of generators of core(I) is 2r+2, where r is the reduction number of I. We prove that this result holds even for ideals which do not have a minimal reduction which is monomial. We also describe the core of an ideal I in k[x,y] having more generators, in the case where I has a monomial minimal reduction. Finally, we consider a strongly stable ideal I in k[x,y] having a monomial minimal reduction J. We prove that r_J(I) leq mu(I)-2, and we give an algorithm for obtaining the core of I via its first coefficient ideal.