Let A be a commutative ring. A graded A-algebra U = ⊕n≥0Un is a standard A-algebra if U0 = A and U = A[U1] is generated as an A-algebra by the elements of U1. A graded U -module F = ⊕n≥0Fn is a standard U -module if F is generated as an U -module by the elements of F0, that is, Fn = UnF0 for all n ≥ 0. In particular, Fn = U1Fn−1 for all n ≥ 1. Given I, J , two ideals of A, we consider the follo...

Theorem. Let R be a Cohen-Macaulay ring (locally, always) 1 c R an ideal o f height at least 2, S the Rees ring of R with respect to I, and G = S /S I the associated graded ring. Assume that S and G are Cohen-Macaulay rings, and that S has a canonical module cos. Then G has a canonical module r and: (i) I f co s can be embedded into S such that cos (considered as an ideal now) is not contained ...

Let k be a field, G an abelian group with a bicharacter, A a colour algebra; i.e., an associative graded k-algebra with identity, C a graded A-coring that is projective as a right A-module, C∗ the graded dual ring of C and Λ a left graded C-comodule that is finitely generated as a graded right C∗-module. We give necessary and sufficient conditions for projectivity and flatness of a graded modul...

New upper bounds on the first and second Hilbert coefficients of a Cohen-Macaulay module over local ring are given. Characterizations provided for some to be attained. The characterizations given in terms series as well Castelnuovo-Mumford regularity associated graded module.

The category of totally ordered graded module braids and that of the exact interlocking sequences are shown to be equivalent. As an application of this equivalence, we show the existence of a connection matrix for a totally ordered graded module braid without assuming the existence of chain complex braid that induces the given graded module braid.

In this paper, we consider the classification of irreducible Zand Z2-graded modules with finite dimensional homogeneous subspaces over the Virasoro-like algebra. We first prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight irreducible modules. As a consequence, we also determine all the modules with nonz...

Given a module M over a ring R that has a grading by a semigroup Q, we present a spectral sequence that computes the local cohomology H I(M) at any graded ideal I in terms of Ext modules. We use this method to obtain finiteness results for the local cohomology of graded modules over semigroup rings. In particular we prove that for a semigroup Q whose saturation Q is simplicial, and a finitely g...