Rigid resolutions and big Betti numbers

Abstract

Abstract. The Betti-numbers of a graded ideal I in a polynomial ring and the Betti-numbers of its generic initial ideal Gin(I) are compared. In characteristic zero it is shown that if these Betti-numbers coincide in some homological degree, then they coincide in all higher homological degrees. We also compare the Betti-numbers of componentwise linear ideals which are contained in each other and have the same Hilbert polynomial. Mathematics Subject Classification (2000). 13D02, 13P10, 13D40, 13A02.

Integrally closed and componentwise linear ideals

“…With the notation of the proof of 2.17, one sees that the assumption (1) can be stated as µ(I) = n−1 i=0 α i (I). Then by [CHH,2.3,1.5], we conclude that I is componentwise linear.…”

“…A sort of Rees property is still valid for m-full ideals not necessarily m-primary. We refer to [CHH,3.2] for the corresponding result for componentwise linear ideals. Proposition 2.16.…”

“…, y p . In order to prove that IJ is componentwise linear, by [CHH,1.5,2.2], it suffices to prove that mH 1 (y 1 , . .…”

“…(3) It follows from [CHH,3.4] that C can be also defined as the class of finite colength ideals I which are componentwise linear with µ(I) = µ(m o(I) ).…”

Rigidity of Linear Strands and Generic Initial Ideals

“…We find the relation between our results for Betti numbers of a graded ideal in a polynomial ring and the Cancellation Principle for generic initial ideals. This paper is organized as follows: In Section 1, we will give an upper bound for graded Betti numbers in terms of generic annihilator numbers by using the technique developed in [10]. In Section 2, we will generalize Conca-Herzog-Hibi's theorem for graded Betti numbers over a polynomial ring.…”