The short exact sequence in definable Galois cohomology

TL;DR

The paper extends definable Galois cohomology to normal extensions in a model-theoretic framework, establishing a short exact sequence that mirrors Serre’s classical and Kolchin’s differential results. It defines definable cocycles and principal homogeneous spaces, and shows that for a normal extension $A\subseteq B$, the cohomology set $H^1_{\mathrm{def}}(B/A,G(B))$ classifies $A$-definable PHSs containing a $B$-point, while an injective map into $H^1_{\mathrm{def}}(M/A,G(M))$ preserves definability. The main theorem then yields a short exact sequence 1 → $H^1_{\mathrm{def}}(B/A,G(B))$ → $H^1_{\mathrm{def}}(C/A,G(C))$ → $H^1_{\mathrm{def}}(C/B,G(C))^{\mathrm{Aut}(B/A)}$, with a transform action ensuring the image lies in fixed points. This work ties together algebraic, differential, and model-theoretic Galois cohomology, enriching the definable-cohomology framework and aligning with prior long-exact sequences.

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 $ω$-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].

Theorems & Definitions (30)

• Proposition 1.1
• Definition 2.1
• Remark 2.2
• Lemma 2.4
• proof
• Proposition 2.5
• proof
• Definition 3.1
• Lemma 3.2
• proof