Abstract. We use the results by Eisenbud and Schreyer [3] to prove that any Betti diagram of a graded module over a standard graded polynomial ring is a positive linear combination Betti diagrams of modules with a pure resolution. This implies the Multiplicity Conjecture of Herzog, Huneke and Srinivasan [5] for modules that are not necessarily Cohen-Macaulay. We give a combinatorial proof of th...

Let Q be an affine semigroup generating Z, and fix a finitely generated Z -graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modulesH I(M) supported on any monomial (that is, Z -graded) ideal...

Let $G$ be a group with identity $e$. $R$ $G$-graded commutative ring and $M$ graded $R$-module. In this paper, we introduce the concept of primary-like submodules as new generalization primary ideals give some basic results about modules. Special attention has been paid, when satisfies gr-primeful property, to find extra properties these submodules.

For any finitely generated torsion-free graded module over a polynomial ring, there exists a homogeneous ideal fitting into an exact sequence similar to a Bourbaki sequence even though its height is not restricted to two.

We introduce the concept of j-stretched ideals in a Noetherian local ring. This notion generalizes to arbitrary ideals the classical notion of stretched m-primary ideals of Sally and RossiValla, as well as the concept of ideals of minimal and almost minimal j-multiplicity introduced by Polini-Xie. One of our main theorems states that, for a j-stretched ideal, the associated graded ring is Cohen...

Given a double complex X there are spectral sequences with the E2 terms being either HI (HII(X)) or HII(HI(X)). But if HI(X) = HII(X) = 0 both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So in this situation the spectral sequences may not give ...

Here we study the maximal dimension of the annihilator ideals 0 :A m j of artinian graded rings A = P/(I, x1, x 2 2, . . . , x 2 v) with a given Hilbert function, where P is the polynomial ring in the variables x1, x2, . . . , xv over a field K with each deg xi = 1, I is a graded ideal of P , and m is the graded maximal ideal of A. As an application to combinatorics, we introduce the notion of ...

The Popescu-Gabriel theorem states that each Grothendieck abelian category is a localization of a module category. In this paper, we prove an analogue where Grothendieck abelian categories are replaced by triangulated categories which are well generated (in the sense of Neeman) and algebraic (in the sense of Keller). The role of module categories is played by derived categories of small differe...