We introduce an idea for generalization of a local cohomology module, which we call a local cohomology module with respect to a pair of ideals (I, J), and study their various properties. Some vanishing and nonvanishing theorems are given for this generalized version of local cohomology. We also discuss its connection with the ordinary local cohomology.

We show that the construction in group cohomology of the Gruenberg resolution associated to a free presentation and the resulting relation module can be copied in the context of representations of categories. We establish five-term exact sequences in the cohomology of categories and go on to show that the Schur multiplier of the category has properties which generalize those of the Schur multip...

the aim of this paper is to study co-prolongations of central extensions. we construct the obstruction theory forco-prolongations and classify the equivalence classes of these by kernels of homomorphisms between 2-dimensional cohomology groups of groups.

EÎ'(X, A) = Ê\T, H\X, A)) where H denotes the Tate cohomology ofir [l, Chapter 12 ] and H'(X, A) is the jth Cech cohomology group of (X, A) with arbitrary coefficients and with w action induced by that on X. The term Ew is the graded module associated with a certain filtered module J*(XyA). This sequence is natural with respect to ir-equivariant maps of (X, A). If w acts trivially on X, the seq...

Complexes constructed from chains of p-subgroups of a group G provide valuable tools in group theory, for the understanding of p-local structure, in algebraic topology, for the study of classifying spaces and mod-p group cohomology through homology decompositions, and in modular representation theory, where the Lefschetz module associated to the complex of all nonidentity p-subgroups can be reg...

For a triple (G,A, κ) (where G is a group, A is a G-module and κ : G → A is a 3-cocycle) and a G-module B we introduce a new cohomology theory 2Hn(G,A, κ;B) which we call the secondary cohomology. We give a construction that associates to a pointed topological space (X, x0) an invariant 2κ 4 ∈ 2H (π1(X), π2(X), κ; π3(X)). This construction can be seen a “3type” generalization of the classical k...

The objective of this paper is to clarify the relationships between quantum Dmodule and equivariant Floer theory. Equivariant Floer theory was introduced by Givental in his paper “Homological Geometry”. He conjectured that the quantum D-module of a symplectic manifold is isomorphic to the equivariant Floer cohomology for the universal cover of the free loop space. First, motivated by the work o...

‎We introduce a generalization of the notion of‎ depth of an ideal on a module by applying the concept of‎ local cohomology modules with respect to a pair‎ ‎of ideals‎. ‎We also introduce the concept of $(I,J)$-Cohen--Macaulay modules as a generalization of concept of Cohen--Macaulay modules‎. ‎These kind of modules are different from Cohen--Macaulay modules‎, as an example shows‎. ‎Also an art...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

It is well known that the homotopy category of connected CW-complexes X whose homotopy groups n;(X) are trivial for i> 1 is equivalent to the category of groups. One of the objects of this paper is to prove a similar equivalence for the connected CW-complexes X whose homotopy groups are trivial for i > n + 1 (where n is a fixed non-negative integer). For n = 1 the notion of crossed module inven...