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.

Let H be a bialgebra. An H-module-algebra is an associative algebra A that is also an H-module such that the multiplication map on A becomes an H-module morphism. This algebraic structure arises often in algebraic topology, quantum groups [11, Chapter V.6], Lie and Hopf algebras theory [3, 16, 19], and group representations [1, Chapter 3]. For example, in algebraic topology, the complex cobordi...

We present short and elementary proofs of two theorems of Huckaba and Marley, while generalizing them at the same time to the case of a module. The theorems concern a characterization of the depth of the associated graded ring of a Cohen-Macaulay module, with respect to a Hilbert filtration, in terms of the Hilbert coefficient e1. As an application, we derive bounds on the higher Hilbert coeffi...

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...

1 Veronese rings, Segre embeddings or more generally Segre-Veronese embeddings are very important rings in Algebraic Geometry. In this paper we present an original, elementary way to compute the Hilbert-Poincare series of these rings, as a consequence we compute their Castelnuovo-Mumford regularity, and also the leading term of the h−vector. Moreover, we can compute the Castelnuovo-Mumford regu...

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...

We define and study the noncommutative spectral flow for paths of regular selfadjoint Fredholm operators on a Hilbert C∗-module. We give an axiomatic description and discuss some applications. One of them is the definition a noncommutative Maslov index for paths of Lagrangians which appears in a splitting formula for the spectral flow. Analogously we study the spectral flow for odd operators on...

In this paper we devote to generalizing some results of componentwise linear modules over a polynomial ring to the ones over a Koszul algebra. Among other things, we show that the i-linear strand of the minimal free resolution of a componentwise linear module is the minimal free resolution of some module which is described explicitly for any i ∈ Z. In addition we present some theorems about whe...