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...

<abstract><p>Let $ R be a G graded commutative ring and M $-graded $-module. The set of all second submodules is denoted by Spec_G^s(M), it called the spectrum $. We discuss rings with Noetherian prime spectrum. In addition, we introduce notion Zariski socle explore their properties. also investigate Spec^s_G(M) topology from viewpoint being space.</p></abstract>

We study cohomology support loci of regular holonomic D-modules on complex abelian varieties, and obtain conditions under which each irreducible component of such a locus contains a torsion point. One case is that both the D-module and the corresponding perverse sheaf are defined over a number field; another case is that the D-module underlies a graded-polarizable mixed Hodge module with a Z-st...

0. Introduction. 1. Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let J C 5 be the subring of all G-invariant polynomials on X. Now let J be the set of all ƒ £ J having zero constant term and ...

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution ...

Let R = ⊕ d∈N0 Rd be a positively graded commutative Noetherian ring which is standard in the sense that R = R0[R1], and set R+ := ⊕ d∈N Rd, the irrelevant ideal of R. (Here, N0 and N denote the set of non-negative and positive integers respectively; Z will denote the set of all integers.) Let M = ⊕ d∈Z Md be a non-zero finitely generated graded R-module. This paper is concerned with the behavi...

For a hyperplane arrangement in real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by Varchenko-Gel'fand ring, which ring functions from chambers to integers with pointwise addition and multiplication. Varchenko Gel'fand gave simple presentation for this along filtration associated graded whose Hilbert series polynomial. We g...

If R is a ring of coefficients and G a finite group, then a flat RG-module which is projective as an R-module is necessarily projective as an RG-module. More generally, if H is a subgroup of finite index in an arbitrary group Γ, then a flat RΓmodule which is projective as an RH-module is necessarily projective as an RΓ-module. This follows from a generalization of the first theorem to modules o...

Let G be a finite group of complex n× n unitary matrices generated by reflections acting on C. Let R be the ring of invariant polynomials, and χ be a multiplicative character of G. Let Ω be the R-module of χ-invariant differential forms. We define a multiplication in Ω and show that under this multiplication Ω has an exterior algebra structure. We also show how to extend the results to vector f...

Irreducible covariant tensor modules for the Lie supergroups GL(m/n) and the Lie superalgebras gl(m/n) and sl(m/n) are obtained through the use of Young tableaux techniques. The starting point is the graded permutation action, first introduced by Dondi and Jarvis, on V. The isomorphism between this group of actions and the symmetric group Sl enables the graded generalization of the Young symmet...