Rigid resolutions and big Betti numbers

Original · EN

In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let I be a homogeneous ideal in a polynomial ring over a field of characteristic 0. Denote by βᵢ(I) the i-th Betti number of I and by Gin(I) the revlex generic initial ideal of I. In general one has βᵢ(I)≤ βᵢ(Gin(I)) and we show that if βᵢ(I)=βᵢ(Gin(I)) for some i then βⱼ(I)=βⱼ(Gin(I)) for all j>i. In the second part of the paper we answer a question of Eisenbud and Huneke. We prove that if I is m-primary and I⊂ mᵈ then βᵢ(mᵈ)≤ βᵢ(Gin(I)) for all i.