We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.Keywords: Simplicial commutative algebra, crossed module of commutative algebras, 2-crossed module of commutative algebras, quadratic derivation.

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.

Abstract:In this study we construct, in the category XAlg(R) /A of crossed A-modules of R-algebroids, the coproduct of given two crossed A-modules M = (µ : M −→ A) and N = (η : N −→ A) of R-algebroids in two di erent ways: Firstly we construct the coproduct M • * N by using the free product M * N of pre-R-algebroids M and N, and then we construct the coproduct M • N by using the semidirect product M N of M and N via µ. Finally we construct an isomorphism between M • * N and M • N.

In this study, we mainly show that the functor from the category X2Mod of 2‐crossed modules of groups to the category Groups of groups assigning to each 2‐crossed module {}L,M,P,∂2,∂1 the group P, and to each 2‐crossed module morphism ()f2,f1,f0, the group homomorphism f0 is a fibration. In addition, we study some related properties.

This paper introduces a categorification of k-algebras called 2-algebras, where k is a commutative ring. We define the 2-algebras as a 2-category with single object in which collections of all 1-morphisms and all 2-morphisms are k-algebras. It is shown that the category of 2-algebras is equivalent to the category of crossed modules in commutative k-algebras.