Adam Van Tuyl: Comparing Invariants of Ideals

Abstract.

Given a homogeneous ideal, there are a number of invariants that can be attached to it, e.g., its regularity or projective dimension. In this talk, I will survey some recent work on comparing invariants. More specifically, given an ideal I that is either an edge ideal or a toric ideal of a graph, I will look at the possible values for the pair \((\deg h_R/I(t), \operatorname{reg}(R/I))\) or \((\operatorname{reg}(R/I), \operatorname{pdim}(R/I))\). Here, \(\deg h_R/I(t)\) is the degree of the \(h\)-polynomial, \(\operatorname{reg}(R/I)\) is the Castelnuovo-Mufmord regularity, and \(\operatorname{pdim}(R/I)\) is the projective dimension.