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Almost cohomology of finite-dimensional Lie rings

Moreno Invitti

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.

Almost cohomology of 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.
Paper Structure (11 sections, 35 theorems, 65 equations)

This paper contains 11 sections, 35 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group theory
  4. Finite-dimensional groups
  5. Lie rings
  6. Almost Lie ring theory
  7. Almost cohomology of Lie rings
  8. Snake Lemma
  9. Almost cohomology groups of a quotient
  10. Finiteness of $\widetilde{H}^1(\mathfrak{g},A)$
  11. Conclusion

Key Result

Theorem 1.4

Let $(\mathfrak{g},A)$ be a module definable in a finite-dimensional theory. Assume that $\mathfrak{g}$ is nilpotent and that $\widetilde{H}^0(\mathfrak{g},A)$ is finite. Then $\widetilde{H}^1(\mathfrak{g},A)$ is finite.

Theorems & Definitions (95)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 85 more