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...

‎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...

This paper is devoted to the horizontal (“characteristic”) cohomology of systems of differential equations. Recent results on computing the horizontal cohomology via the compatibility complex are generalized. New results on the Vinogradov C-spectral sequence and Krasilshchik’s C-cohomology are obtained. As an application of general theory, the examples of an evolution equation and a p-form gaug...

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural generalization of Deligne cohomology. For smooth complex algebraic varieties, we show that the theory satisfies a projective bundle formula and A1-homotopy invarianc...

It is well known that the validity of the so called LenardMagri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson coho...