A graded tensor category over a group G will be called a strongly G-graded tensor category if every homogeneous component has at least one invertible object. Our main result is a description of the module categories over a strongly G-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of G.

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with a non-zero identity and $M$ be a graded $R$-module. In this article, we introduce the concept of graded almost semiprime submodules. Also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

Let R be a G-graded ring and M be a G-graded R-module. In this article, we introduce the concept of graded weakly classical prime submodules and give some properties of such submodules.

Let A be a (G, χ)-Hopf algebra with bijection antipode and let M be a G-graded A-bimodule. We prove that there exists an isomorphism HH∗gr(A,M) ∼= Ext ∗ A-gr(K, (M)), where K is viewed as the trivial graded A-module via the counit of A, M is the adjoint A-module associated to the graded A-bimodule M and HHgr denotes the G-graded Hochschild cohomology. As an application, we deduce that the cohom...

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 G be a group with identity e. Let R be a G-graded commutative ring and let M be a graded R-module. The graded classical prime spectrum Cl.Specg(M) is defined to be the set of all graded classical prime submodule of M. The Zariski topology on Cl.Specg(M); denoted by ϱg. In this paper we establish necessary and sufficient conditions for Cl.Specg(M) with the Zariski topology to be a Noetherian...

We introduce a notion of strongly C×-graded, or equivalently, C/Zgraded generalized g-twisted V -module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of strongly C-graded generalized g-twisted V -module if V admits an additional C-grading compatible with g. Let V = ∐ n∈Z V(n) be a vertex operator algebra such that V(0)...

For a coherent filtered D-module we show that the dual of each graded piece over the structure sheaf is isomorphic to a certain graded piece of the ring-theoretic local cohomology complex of the graded quotient of the dual of the filtered D-module along the zero-section of the cotangent bundle. This follows from a similar assertion for coherent graded modules over a polynomial algebra over the ...

The graded Specht module S for a cyclotomic Hecke algebra comes with a distinguished generating vector z ∈ S, which can be thought of as a “highest weight vector of weight λ”. This paper describes the defining relations for the Specht module S as a graded module generated by z. The first three relations say precisely what it means for z to be a highest weight vector of weight λ. The remaining r...

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...