Regularity of Tor, LCM-Duals, and Hilbert Functions

In this dissertation, we study several problems in commutative algebra. The first problem we study is the regularity of Tor for weakly stable ideals. We prove that if I and J are weakly stable ideals in a polynomial ring R=k[x_1,...,x_n] over a field k, then the regularity of Tor^R_i(R/I,R/J) is bounded by reg^R R/I + reg_R R/J + i. We also give a bound for the regularity of Ext^i_R(R/I,R) for I a weakly stable ideal.

For the second problem, we define an operation on monomial ideals known as the LCM-dual. We study the properties of the LCM-dual for several classes of monomial ideals. The first class of ideals that arises from graph theory are Ferrers ideals, which are edge ideals of a Ferrers graph. We show that the LCM-dual of a Ferrers ideal is the Alexander dual of the edge ideal of the complement of the Ferrers graph. The second class of ideals we consider are specializations of Ferrers ideals, strongly stable ideals of degree two. We find a cellular complex which supports the minimal free resolution for the LCM-duals of these strongly stable ideals. We describe minimal free resolutions of LCM-duals of strongly stable ideals generated in degree two. We describe the special fiber ring in this case. Using the same technique, we consider a larger class of ideals generated in degree two, and find their minimal free resolutions. We also show that when the height of a monomial ideal I is at least 2, the special fiber ring of I is isomorphic to the special fiber ring of the LCM-dual of I. We use this to describe the special fiber rings of LCM-duals of strongly stable ideals in degree two.

The third problem we consider deals with Hilbert functions of Cohen-Macaulay local rings. Given a Cohen-Macaulay local ring (R, m) of dimension d > 0, we study the depth of the associated graded ring of R with respect to the maximal ideal m. We analyze the case where the multiplicity e is h+3, where h is the embedding codimension. The tool we use is the bigraded Sally module S of m with respect to J, where J is a minimal reduction of m.