Abstract. We study h-vectors and graded Betti numbers of level modules up to multiplication by a rational number. Assuming a conjecture on the possible graded Betti numbers of Cohen-Macaulaymodules we get a description of the possible h-vectors of level modules up to multiplication by a rational number. We also determine, again up to multiplication by a rational number, the cancellable h-vector...

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of Artinian Gorenstein algebras with the Weak Lefschetz Property, a property shared by most Artinian Gorenstein algebras. Starting with an arbitrary SI-sequence, we construct a reduced, arithmeti...

Abstract. Let I = (F1, . . . , Fr) be a homogeneous ideal of the ring R = k[x0, . . . , xn] generated by a regular sequence of type (d1, . . . , dr). We give an elementary proof for an explicit description of the graded Betti numbers of Is for any s ≥ 1. These numbers depend only upon the type and s. We then use this description to: (1) write HR/Is , the Hilbert function of R/Is, in terms of HR...

In a remarkable paper Mats Boij and Jonas Söderberg [2006] conjectured that the Betti table of a Cohen-Macaulay module over a polynomial ring is a positive linear combination of Betti tables of modules with pure resolutions. We prove a strengthened form of their Conjectures. Applications include a proof of the Multiplicity Conjecture of Huneke and Srinivasan and a proof of the convexity of a fa...

Let P be a finite partially ordered set with unique minimal element 0̂. We study the Betti poset of P , created by deleting elements q ∈ P for which the open interval (0̂, q) is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form (0̂, p) ⊂ P and corresponding open intervals in the Betti poset. Our motivating application is that the...

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...

We first describe a situation in which every graded Betti number in the tail of the resolution of R J may be read from the socle degrees of R J . Then we apply the above result to the ideals J and J [q]; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R/J [q] are equal to the graded Betti numbers in the tail of a shift of the resolution of R/J...

It gives a class of p-Borel principal ideals of a polynomial algebra over a field K for which the graded Betti numbers do not depend on the characteristic of K and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal p-Borel ideals have binomial cycle basis on Koszul homology modules.

In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.

Under reasonable assumptions, a group action on module extends to the minimal free resolutions of module. Explicit descriptions these actions can lead better understanding by providing, for example, convenient expressions their differentials or alternative characterizations Betti numbers. This article introduces an algorithm computing characters finite groups acting finitely generated graded mo...