Fellowship of the Ring, National Seminar: Subadditivity of Syzygies and Related Problems
Jason McCullough (Iowa State University)
To attend this seminar, you must register in advance, by clicking HERE.
Let S = K[x_1,...,x_n] be a polynomial ring over a field and I a graded S-ideal. There are many interesting questions about the maximal graded shifts of S/I, denoted t_i. In the first part of my talk, I will discuss two classical constructions that turn a (graded) S-module into an ideal with similar properties, namely idealizations and Bourbaki ideals, and what they say about maximal graded shifts of ideals. In the second part of the talk, I will discuss restrictions on maximal graded shifts of ideals. In particular, an ideal I is said to satisfy the subadditivity condition if t_a + t_b ≥ t_(a+b) for all a,b. This condition fails for arbitrary, even Cohen-Macaulay, ideals but is open for certain nice classes of ideals, such as Koszul and monomial ideals. I will present a construction (joint with A. Seceleanu) showing that subadditivity can fail for Gorenstein ideals.
If time allows, I will talk about some results that hold more generally, including a linear bound on the maximal graded shifts in terms of the first p-c shifts, where p = pd(S/I) and c = codim(I). I hope to include several examples and open questions as well.