Seminar

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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.

Notes 9.81 MB application/pdf

Subadditivity Of Syzygies And Related Problems