Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, Vasconcelos shows that both can be compared in case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained i...

We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are...

We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad, and on the example of commutative alegbras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these cat...

The study of Koszul algebras and their representations has accelerated significantly in the last few years. They have been used in Topology, Algebraic Geometry and Commutative Algebra and they are used more and more frequently in Representation Theory, see for instance [BGS],[GTM],[M],[MZ] and [R]. The aim of this paper is to present some of the results presented by the second author at the Lum...

Suppose S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, E is a coherent OX -bimodule and I ⊂ T (E) is a graded ideal. We study the geometry of the functor Γn of flat families of truncated B = T (E)/I-point modules of length n + 1. We then use the results of our study to show that the point modules over B are parameterized by the closed points of P X2(E). When X = P , w...

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded k-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically CohenMacaulay one-dimensional schemes of ...

We introduce group corings, and study functors between categories of comodules over group corings, and the relationship to graded modules over graded rings. Galois group corings are defined, and a Structure Theorem for the G-comodules over a Galois group coring is given. We study (graded) Morita contexts associated to a group coring. Our theory is applied to group corings associated to a comodu...

The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded k-algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-s...

Let $R = bigoplus_{n in mathbb{N}_{0}} R_{n}$ be a standardgraded ring, $M$ be a finitely generated graded $R$-module and $J$be a homogenous ideal of $R$. In this paper we study the gradedstructure of the $i$-th local cohomology module of $M$ defined by apair of ideals $(R_{+},J)$, i.e. $H^{i}_{R_{+},J}(M)$. Moreprecisely, we discuss finiteness property and vanishing of thegraded components $H^...

We prove that the Auslander-Reiten conjecture holds for commutative standard graded artinian algebras, in two situations: The first is under the assumption that the modules considered are graded and generated in a single degree. The second is under the assumption that the algebra is generic Gorenstein of socle degree 3.