A polynomial bound on the regularity of an ideal in terms of half of the syzygies

Let $K$ be a field and let $S = K[x_1,\ldots,x_n]$ be apolynomial ring. Consider a homogenous ideal $I \subsetS$. Let $t_i$ denote $\reg(\tor_i^S(S/I,K))$, the maximaldegree of an $i$th syzygy of $S/I$. We prove bounds on thenumbers $t_i$ for $i > \lceil \frac{n}{2} \rceil$ purely interms of the previous $t_i$. As a result, we give boundson the regularity of $S/I$ in terms of as few as half ofthe numbers $t_i$. We also prove related bounds forarbitrary modules. These bounds are often much smallerthan the known doubly exponential bound on regularitypurely in terms of $t_1$.

Keywords

regularity, Betti numbers, resolution

2010 Mathematics Subject Classification

Primary 13D02. Secondary 13D07, 13F20.

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Published 8 November 2012