A Generalization of the Taylor Complex Construction

Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals. Introduction Let K be a field, S = K[x1, . . . , xn] the polynomial ring in n variables over K, and let I and J be two monomial ideals in S. Suppose that F is a multigraded free S-resolution of S/I and G a multigraded free S-resolution of S/J . In this note we construct a multigraded free resolution of S/(I + J) which we denote by F ∗ G. It follows from our construction that βi(S/(I + J)) ≤ ∑i j=0 βj(S/I)βi−j(S/J) for all i ≥ 0. Here βi(M) denotes the ith Betti number of a graded S-module M , that is, the K-dimension of TorSi (K,M). The inequality for the Betti-numbers implies in particular that proj dim(I +J) ≤ proj dim(I) + proj dim(J) + 1. The numerical data of the complex F ∗ G also yield the inequality reg(I + J) ≤ reg(I) + reg(J) − 1. Similar inequalities hold for the projective dimension and the regularity of I ∩ J , see Section 3. The inequality for the regularity has first been conjectured by Terai [7]. He also proved this inequality in a special case. In the squarefree case these inequalities have first been proved by Kalai and Meshulam [5]. The construction of the complex F ∗ G was inspired by the work of Kalai and Meshulam. In fact, the first author informed me that the above mentioned inequalities for the projective dimension and the regularity of sums and intersections of squarefree monomial ideals follow from certain inequalities proved in [5] concerning the d-Leray properties of the union and intersection of simplicial complexes. Thus our construction provides an algebraic explanation of these inequalities. One should note that for example the inequality proj dim(I + J) ≤ proj dim(I) + proj dim(J) + 1, as well as all the other inequalities, are wrong for arbitrary graded ideals. We would also like to mention that the Taylor resolution (cf.[4]) is a special case of our construction. The Taylor resolution is a multigraded free resolution for monomial ideals. It has a uniform structure, but in most cases, the Taylor resolution is nonminimal. In the frame of our construction the Taylor resolution can be described as follows: if I ⊂ S is a monomial ideal with the minimal set of monomial generators 1 {u1, . . . , ur}, and Fj is the graded minimal free resolution of the principal ideal (uj) for j = 1, . . . , r, then F1 ∗ F2 ∗ · · · ∗ Fr is the Taylor resolution of S/I. 1. The construction Let K be field, S = K[x1, . . . , xn] a polynomial ring and I ⊂ S a monomial ideal. Then S/I admits a multigraded minimal free S-resolution F : 0 −−−→ Fp φp −−−→ Fp−1 φp−1 −−−→ · · · φ2 −−−→ F1 φ1 −−−→ F0 −−−→ 0, that is, one has (i) H0(F) = S/I; (ii) Fi = ⊕ j S(−aij) with aij ∈ Z n for all i; (iii) the differentials φi are homomorphisms of multigraded modules. We define a partial order on Z by saying that b ≤ a for a, b ∈ Z, if b is componentwise less than a. For all i let Bi be a multihomogeneous basis of Fi. Then Fi = ⊕ g∈Bi Sg, and the differential φi : Fi → Fi−1 can be described by the equations φi(g) = ∑