In this paper we investigate the relationship between simplicial and crossed resolutions of commutative algebras.

In this paper, we described the GAP implementation of crossed modules of commutative algebras and cat 1 -algebras and their equivalence. We include a table of cat 1 -structures on algebras

We introduce the notion of a 3-crossed module, which extends the notions of a 1-crossed module (Whitehead) and a 2-crossed module (Conduché). We show that the category of 3-crossed modules is equivalent to the category of simplicial groups having a Moore complex of length 3. We make explicit the relationship with the cat 3 -groups (Loday) and the 3-hypercomplexes (Cegarra-Carrasco), which also model algebraically homotopy 4-types.

In this paper a definition of a category of modules over the ring of differential operators on a smooth variety of finite type in positive characteristics is given. It has some of the good properties of holonomic D-modules in zero characteristic. We prove that it is a Serre category and that it is closed under the usual D-module functors, as defined by Haastert. The relation to the similar concept of F-finite modules, introduced by Lyubeznik, is elucidated, and several examples, such as etale algebras, are given.

In this work, we define the quadratic modules for commutative algebras and give relations among 2-crossed modules, crossed squares, quadratic modules and simplicial commutative algebras with Moore complex of length 2.