Yoshizawa investigated when local cohomology modules have an annihilator that does not depend on the choice of defining ideal. In this paper we refine his results and investigate relationship between annihilators restricted flat dimensions.

In this paper we present algorithms that compute certain local cohomology modules associated to a ring of polynomials containing the rational numbers. In particular we are able to compute the local cohomological dimension of algebraic varieties in characteristic zero.

For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology.

We introduce the notion of 3-Hom-Lie-Rinehart algebras and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions algebra characterize first space in terms group automorphisms an A-split abelian extension equivalence classes extensions. Finally, study formal deformations algebras.

We continue studying the class of modules having reducible complexity over a local ring. In particular, a method is provided for computing an upper bound of the complexity of such a module, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which all modules have finite complexity.