The short exact sequence in definable Galois cohomology

TL;DR

This paper proves that a short exact sequence of automorphism groups in a model-theoretic setting induces a corresponding short exact sequence in definable Galois cohomology, extending the understanding of cohomological structures in model theory.

Contribution

It demonstrates that automorphism group sequences induce exact sequences in definable Galois cohomology within a model-theoretic framework, expanding prior results.

Findings

Short exact sequences of automorphism groups induce exact sequences in definable Galois cohomology.

The results complement existing long exact sequences in definable Galois cohomology.

The work discusses compatibilities with previous cohomological frameworks.

Abstract

In Remarks on Galois Cohomology and Definability [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let $M$ be an atomic and strongly $\omega$-homogeneous structure over a set of parameters $A$. Let $B$ be a normal extension of $A$ in $M$. We show that a short exact sequence of automorphism groups $1 \to \text{Aut}(M/B) \to \text{Aut}(M/A) \to \text{Aut}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in More on Galois cohomology, definability and differential algebraic groups [4].