Almost cohomology of finite-dimensional Lie rings

TL;DR

This paper introduces the concept of almost cohomology groups for finite-dimensional Lie rings, establishing their properties and finiteness conditions, which extend classical cohomology theory in this algebraic context.

Contribution

It defines almost cohomology groups for Lie rings and proves finiteness results linking the 0th and 1st groups, advancing the understanding of cohomological properties in Lie ring theory.

Findings

Defined 0th and 1st almost cohomology groups for Lie rings.

Proved the 1st almost cohomology group is finite if the 0th is finite.

Extended classical cohomology concepts to finite-dimensional Lie rings.

Abstract

We introduce almost cohomology groups for Lie rings definable in finite-dimensional theory. In particular, we define the 0th and 1st almost cohomology groups of a Lie ring module. Moreover, we prove that the 1st almost cohomology group of a finite-dimensional definable Lie ring module is finite if the 0th almost cohomology group is finite.