. A C ] 1 6 Ju n 20 03 RIGID RESOLUTIONS AND BIG BETTI NUMBERS

Let K be a field of characteristic 0 and S = K[x1, . . . , xn] the polynomial ring over K with graded maximal ideal m = (x1, . . . , xn). Denote by βi(M) = Tor S i (K,M) the ith Betti number of a finitely generated graded module M and by Gin(I) the generic initial ideal of a graded ideal I with respect to the reverse lexicographical order. In this paper we answer (positively) a question raised by the first author in [7]. We prove that if a graded ideal I ⊂ S has βi(I) = βi(Gin(I)) for some i, then βk(I) = βk(Gin(I)) for all k ≥ i, see Corollary 2.4. For i = 0, this theorem was first proved by Aramova, Herzog and Hibi [1]. More generally, we show that the same statement holds if Gin(I) is replaced by either any generic initial ideal of I or by the lex-segment ideal associated with I. Given a finitely generated graded S-module M , a generic sequence of linear forms y1, . . . , yn and an integer p, 1 ≤ p ≤ n, we define the generic annihilator number αp(M) of M to be dimK((y1, . . . , yp−1)M :M yp/(y1, . . . , yp−1)M and the generic Koszul homology Hi(y1, . . . , yp;M) to be the ith-homology of the Koszul complex over M with respect to y1, . . . , yp. In the first section of this paper we show that there is an upper bound for the Betti numbers of M in terms of the generic annihilator numbers of M . We show in Theorem 1.5, that among other equivalent conditions, this upper bound is achieved for all i if and only if mHi(y1, . . . , yp;M) = 0 for all i > 0 and p = 1, . . . , n. The above mentioned Corollary 2.4 is a consequence of the more general Theorem 2.3, proved in Section 2, which says that if the ith Betti number of M achieves the upper bound given by the generic annihilator numbers, then the upper bound is also achieved for jth Betti numbers with j > i. For the proof of this theorem the following interesting annihilation property of Koszul homology is required: suppose that for a generic sequence y = y1, . . . , yn of linear forms and some i one has mHi(y1, . . . , yp;M) = 0 for all p then mHk(y1, . . . , yp;M) = 0 for all k ≥ i and for all p. In the last section of the paper we show that if two componentwise linear ideals I ⊂ J ⊂ S have the same Hilbert polynomial then βi(J) ≤ βi(I) for all i, see Theorem 3.2. This theorem was inspired by a question of Eisenbud and Huneke: suppose char(K) = 0 and I is a graded m-primary ideal in S with I ⊂ m for some d. Is it then true that the number of generators of Gin(I) is greater than or equal to the number of generators of m? As an application of Theorem 3.2 we show in Corollary 3.4 that this is indeed the case. Moreover, we show that Gin(I) and m