The Grothendieck-Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two m-primary ideals I and J in a local ring (A, m) in terms of local cohomology modules of Rees algebras ...

All rings in this paper are commutative and Noetherian. If R is a ring and I ⊂ R is an ideal, cd(R, I) denotes the cohomological dimension of I in R, i.e. the largest integer i such that the i-th local cohomology module H i I(M) doesn’t vanish for some R-module M . For the purposes of this introduction R is a complete equicharacteristic regular local d-dimensional ring with a separably closed r...

Let R = ⊕ n≥0 Rn be a homogeneous Noetherian ring with local base ring (R0,m0) and let M be a finitely generated graded R-module. Let Hi R+ (M) be the i-th local cohomology module of M with respect to R+ := ⊕ n>0 Rn. If dimR0 ≤ 1, the R-modules Γm0R(H R+ (M)), (0 :Hi R+ (M) m0) and Hi R+ (M)/m0H i R+ (M) are Artinian for all i ∈ N0. As a consequence, much can be said on the asymptotic behaviour...

In the definition of a crossed module $(T,G,rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate som...

The cohomology of a group G with coefficients in a left G–module M is denoted H .GIM /. We are primarily interested in the case where M is the group ring, ZG . Since ZG is a G–bimodule, H .GIZG/ inherits the structure of a right G– module. When G is discrete and acts properly and cocompactly on a contractible CW complex , there is a natural topological interpretation for this cohomology group:...

We generalize Griffiths’ theorem on the Hodge filtration of the primitive cohomology of a smooth projective hypersurface, using the local Bernstein-Sato polynomials, the V -filtration of Kashiwara and Malgrange along the hypersurface and the Brieskorn module of the global defining equation of the hypersurface.