Simplicial and crossed Lie algebras

Abstract

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.

Peiffer elements in simplicial groups and algebras

“…They have also shown the equality for n = 2, 3 and 4, and argued for its validity for all n. A similar result for simplicial Lie algebras was obtained by Arvasi and Akça in [1].…”

“…The objective of this paper is to give a general proof for the inclusions partially proved in [1,3,15] and [16]. The methods used by the previous authors to study higher-dimensional Peiffer elements are based on the ideas of Conduché [9] and the techniques developed by Carrasco and Cegarra [6,7].…”

“…The decomposition of the group of boundaries of the Moore complex of a simplicial group (algebra) as a product of commutator subgroups (sum of product ideals) is of interest in various topological and homological settings. For instance, in calculations of non-abelian homology of groups [12] and non-abelian homology of Lie algebras [13], and to explain the relations among several algebraic models of connected homotopy 3-types: braided regular crossed modules, 2-crossed modules, quadratic modules, crossed squares and simplicial groups with Moore complex of length 2 [1,3,15,16]. It is possible that such decomposition contributes to light complete descriptions of algebraic models of the n-types of specific families of spaces for low values of n, to calculate Samelson and Whitehead products [10] and analogues in homotopy theory of simplicial Lie algebras or to link simplicial groups and weak infinity categoric models.…”

3-types of simplicial groups and braided regular crossed modules

“…Mutlu and Porter [24] have also adapted their method to simplicial groups. Castiglioni and Ladra [15] generalized the results proved in [1], [4] and [24].…”

“…Castiglioni and Ladra [15] gave a general proof for the inclusions partially proved by Arvasi and Porter in [4], Arvasi and Akça in [1] and Mutlu and Porter in [24]. Their approach to the problem is different from that of the cited works.…”