Stabilization of Betti tables

Abstract

Abstract. Let I ⊆ R = k[x 1 , ..., xn] be a homogeneous equigenerated ideal of degree r. We show here that the shapes of the Betti tables of the ideals I d stabilize, in the sense that there exists some D such that for all d ≥ D, β i,j+rd (I d ) = 0 ⇔ β i,j+rD (I D ) = 0. We also produce upper bounds for the stabilization index Stab(I). This strengthens the result of Cutkosky, Herzog, and Trung that the Castelnuovo-Mumford regularity reg(I d ) is eventually a linear function in d.

Virtual resolutions of monomial ideals on toric varieties

“…This result is similar in flavor to results of Mayes-Tang [MT19] and Whieldon [Whi14], which describe the stabilization of the shapes and decompositions of the betti table of the usual power of an ideal.…”

Stabilization of Boij–Söderberg decompositions of ideal powers

“…The following result states that the shape formed by the nonzero entries in the Betti tables β(I k ) stabilize as k gets large. It appears to have been proven independently by Lavila-Vidal, Singla, and Whieldon, [Sin07], [Whi14]). Let I be an ideal of the a polynomial ring S that is equigenerated in degree r. Then there exists a k 0 such that for all k > k 0 , ).…”

“…Given an ideal I, we expect that the generating set, Betti numbers, and thus corresponding Boij-Söderberg decompositions of its powers I k will become increasingly complicated as we increase k. Indeed, this is what usually happens. However, we restrict our attention to ideals I generated by homogeneous polynomials of the same degree, there is a stabilization in the Betti numbers ( [LV04], [Sin07], [Whi14]). Engström conjectured that, when I is generated by monomials of the same degree, there is a corresponding stabilization in the Boij-Söderberg decompositions ([Eng13]).…”

The eventual shape of Betti tables of powers of ideals

“…While writing this paper, we were informed by Whieldon that in her recent work [16], a similar result to Corollary 1.4 (1)- (3) is proved. We later learned that Pooja Singla also proved these results, in the second chapter of her thesis [14], independently.…”