Ideals with Stable Betti Numbers

✍ Scribed by Annetta Aramova; Jürgen Herzog; Takayuki Hibi

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 91 KB
• Volume: 152
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1888

✦ Synopsis

Componentwise linear ideals were introduced earlier to generalize the result that the Stanley Reisner ideal I 2 of a simplicial complex 2 has a linear resolution if and only if its Alexander dual 2* is Cohen Macaulay. It turns out that I 2 is componentwise linear if and only if 2* 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.

2000 Academic Press

1. GINS AND COMPONENTWISE LINEAR IDEALS

Let K be a field of characteristic 0 and S=K[x 1 , ..., x n ] the polynomial ring over K with each deg x i =1. We work with the reverse lexicographic term order on S induced by x 1 > } } } >x n . For a graded ideal I of S, let Gin(I) denote the generic initial ideal of I with respect to this term order, see, e.g., [8,10].

If I is a graded ideal of S, then we write I ( j ) for the ideal generated by all homogeneous polynomials of degree j belonging to I. Moreover, we write I d for the ideal generated by all homogeneous polynomials of I whose degree is greater than or equal to d. We say that a graded ideal I/S