Subadditivity of syzygies and related problems

Abstract: 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.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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