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

Abstract

Abstract. Let K be a field and let S = K[x 1 , . . . , x n ] be a polynomial ring. Consider a homogenous ideal I ⊂ S. Let t i denote reg(Tor S i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t i for i > n 2 purely in terms of the previous t i . As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t i . We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t 1 .

On the maximal graded shifts of ideals and modules

“…By adapting an argument of Herzog-Srinivasas [14], we also prove in Theorem 3.1 the special case of Conjecture 1.4 when J is principal. This recovers and strengthens results of Herzog-Srinivasan [14] and El Khoury-Srinivasan [11] as well as the author [17].…”

“…Note that if Conjecture 1.4 is true, then for any homogeneous ideal I with pd(S/I) ≤ 2 codim(I), Question 5.1 in [17] has an affirmative answer.…”

“…Following the notation in [11], we denote the ith maximal and minimal graded shifts as T i (M ) = max{j | Tor S i (M, k) j = 0} = max{j | β ij (M ) = 0}, t i (M ) = min{j | Tor S i (M, k) j = 0} = min{j | β ij (M ) = 0}. Other authors [2,14,17] have used t i (M ) to denote the maximal shifts but the above notation seems more apt to the situation of dealing with both maximal and minimal shifts. The maximal shifts are of primary interest due to their connection with Castelnuovo-Mumford regularity, that is…”

“…, T h (S/I), where h := p 2 . In [17,Theorem 4.7] the author proved a weaker polynomial bound on reg(S/I), and in particular on T p (S/I), in terms of T 1 (S/I), . .…”

Bounds on the regularity and projective dimension of ideals associated to graphs

“…However, the known examples of ideals having high regularity have the property that "most" of the contributions to such high regularity occur at the beginning of the resolution. (See the recent preprint [23] for a detailed discussion and results on this matter.) The general philosophy, then, is that beginning syzygies have "most" of the regularity in them.…”