‎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 Q be an affine semigroup generating Z, and fix a finitely generated Z -graded module M over the semigroup algebra k[Q] for a field k. We provide an algorithm to compute a minimal Z-graded injective resolution of M up to any desired cohomological degree. As an application, we derive an algorithm computing the local cohomology modulesH I(M) supported on any monomial (that is, Z -graded) ideal...

This paper finds its motivation in the pursuit of ideals whose local cohomology modules have maximal Hilbert functions. In [8], [9] we proved that the lexicographic (resp. squarefree lexicographic) ideal of a family of graded (resp. squarefree) ideals with assigned Hilbert function provides sharp upper bounds for the local cohomology modules of any of the ideals of the family. Moreover these bo...

We deene two (n + 1) graded Lie brackets on spaces of multilinear mappings. The rst one is able to recognize n-graded as-sociative algebras and their modules and gives immediately the correct diierential for Hochschild cohomology. The second one recognizes n-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.

We define two (n + 1) graded Lie brackets on spaces of multilinear mappings. The first one is able to recognize n-graded as-sociative algebras and their modules and gives immediately the correct differential for Hochschild cohomology. The second one recognizes n-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.