Linear Components and the Behavior of Graded Betti Numbers under the Transition to Generic Initial Ideals

Let K be a field, S a polynomial ring and E an exterior algebra over K, both in a finite set of variables. In this paper we study the graded Betti numbers of graded ideals in S and E. First, we prove that if the graded Betti number β ii+k(S/I) = β S ii+k(S/Gin(I)) for some i > 1 and k ≥ 0 then one has β qq+k(S/I) = β S qq+k(S/Gin(I)) for all q ≥ i, where I ⊂ S is a graded ideal. Second, we show that if β ii+k(E/I) = β E ii+k(E/Gin(I)) for some i > 1 and k ≥ 0 then one has β qq+k(E/I) = β E qq+k(E/Gin(I)) for all q ≥ 1, where I ⊂ E is a graded ideal. In addition, it will be shown that β ii+k(R/I) = β R ii+k(R/Gin(I)) for all i ≥ 1 if and only if I〈k〉 and I〈k+1〉 have a linear resolution. Here I is the ideal generated by all homogeneous elements in I of degree d, and R can be either the polynomial ring or the exterior algebra. Introduction In 2004, Conca, Herzog and Hibi [10] proved the following attractive theorem: Let S = K[x1, . . . , xn] be the polynomial ring over a field K with char(K) = 0, I ⊂ S a graded ideal, Gin(I) the generic initial ideal of I with respect to the reverse lexicographic order induced by x1 > · · · > xn and β S i (S/I) the i-th total Betti number of S/I over S. They proved that if β i (S/I) = β S i (S/Gin(I)) for some i ≥ 1 then we have β q (S/I) = β S q (S/Gin(I)) for all q ≥ i. In this paper, we will extend Conca–Herzog–Hibi’s theorem and prove related theorems for the graded Betti numbers of graded ideals in the polynomial ring and in the exterior algebra. First, we will consider the polynomial ring. Let β ij(M) be the i, j-th graded Betti number of a finitely generated graded S-module M over S. What we prove is the following: Let K be a field of characteristic 0 and I a graded ideal of S. If β ii+k(S/I) = β S ii+k(S/Gin(I)) for some i > 1 and k ≥ 0, then we have β qq+k(S/I) = β S qq+k(S/Gin(I)) for all q ≥ i. Next, we will study the same property for generic initial ideals over the exterior algebra. Let K be an infinite field, V an n-dimensional K-vector space with basis e1, . . . , en and E = ⊕n k=0 ∧k V the exterior algebra of V . For a graded ideal J ⊂ E, we write Gin(J) for the generic initial ideal of J with respect to the reverse lexicographic order induced by e1 > · · · > en and β E ij (E/J) the i, j-th graded Betti number of E/J over E. Somewhat surprisingly, the following stronger property is 2000 Mathematics Subject Classification. 13D02. The first author is supported by JSPS Research Fellowships for Young Scientists.