Ideals with Stable Betti Numbers

Abstract

Componentwise linear ideals were introduced earlier to generalize the result that the Stanley–Reisner ideal IΔ of a simplicial complex Δ has a linear resolution if and only if its Alexander dual Δ* is Cohen–Macaulay. It turns out that IΔ is componentwise linear if and only if Δ* is sequentially Cohen–Macaulay. In this paper we discuss Betti number properties of componentwise linear ideals. Let I be a graded ideal of a polynomial ring S and Gin(I) the generic initial ideal of I with respect to the reverse lexicographic term order on S. Our main result is that I and Gin(I) admit the same graded Betti numbers if and only if I is componentwise linear. For the proof of this fact, we describe some properties of the Betti diagram of a generic initial ideal. Combinatorial implications for shifted complexes will also be discussed.