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...

In the spirit of noncommutative geometry we construct all inequivalent vector bundles over the (2, 2)-dimensional supersphere S2,2 by means of global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding ‘rank 1’ supervector bundle over S2,2. The canonical connection ∇ = p ◦ d is used to compute the Chern numbers by me...

We provide a uniform vanishing result for the graded components of finite length Koszul module associated to subspace K⊆∧2V as well sharp upper bound its Hilbert function. This purely algebraic statement has interesting applications study number invariants finitely generated groups, such Alexander invariants, Chen ranks, and degree growth virtual nilpotency class. For instance, we explicitly af...

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...

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...

We show that  higher derivations on a Hilbert$C^{*}-$module associated with the Cauchy functional equation satisfying generalized Hyers--Ulam stability.