3-Crossed Modules of Commutative Algebras

TL;DR

This paper introduces 3-crossed modules for commutative algebras, explores their relation to simplicial algebras, and constructs a projective resolution to study higher-dimensional homological properties.

Contribution

It defines 3-crossed modules for commutative algebras, establishes their connection with simplicial structures, and proves the existence of a projective 3-crossed resolution for any algebra.

Findings

Established the definition of 3-crossed modules for commutative algebras

Linked 3-crossed modules with simplicial algebra structures

Proved the existence of a projective 3-crossed resolution for arbitrary algebras

Abstract

In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher dimensional homological information and show the existence of this resolution for an arbitrary $\mathbf{k}$-algebra.