We find a relationship between the graded quotients of a filtered holonomic D-module, their duals as coherent sheaves, and the characteristic variety, in case the filtered D-module underlies a polarized Hodge module on a smooth algebraic variety. The proof is based on M. Saito’s result that the associated graded module is Cohen–Macaulay, and on local duality for the cotangent bundle. The result...

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...

We study Gorenstein dimension and grade of a module M over a filtered ring whose assosiated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G-dimM ≤ G-dimgrM and an equality gradeM = grade grM , whe...

Let Δ be an abelian group. By considering the notion multiplication of Δ-graded modules (see [7]) over a commutative Δ-graded ring with unity, we introduce the notion of product of two Δ-graded submodules which we use to characterize the Δ-graded prime submodules of a multiplication Δ-graded module. Finally we proved a graded version of Nakayama lemma for multiplication Δ-graded modules.

We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.

Let A be a Z-graded associative algebra and let ρ be an irreducible N-graded representation of A on W with finite-dimensional homogeneous subspaces. Then it is proved that ρ(Ã) = glJ (W ), where à is the completion of A with respect to a certain topology and glJ (W ) is the subalgebra of EndW , generated by homogeneous endomorphisms. It is also proved that an N-graded vector space W with finite...

Let $R$ be a $G$-graded ring and M $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to set all submodules $Q$ such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we topology on $\mathcal{PS}_G(M)$ having Zariski prime $Spec_G(M)$ as subspace topology, investigate several topological properties space.

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this ...