{"title":"The short exact sequence in definable Galois cohomology","authors":"David Meretzky","doi":"arxiv-2408.04147","DOIUrl":null,"url":null,"abstract":"In Remarks on Galois Cohomology and Definability [2], Pillay introduced\ndefinable Galois cohomology, a model-theoretic generalization of Galois\ncohomology. Let $M$ be an atomic and strongly $\\omega$-homogeneous structure\nover a set of parameters $A$. Let $B$ be a normal extension of $A$ in $M$. We\nshow that a short exact sequence of automorphism groups $1 \\to \\text{Aut}(M/B)\n\\to \\text{Aut}(M/A) \\to \\text{Aut}(B/A) \\to 1$ induces a short exact sequence\nin definable Galois cohomology. Our result complements the long exact sequence\nin definable Galois cohomology developed in More on Galois cohomology,\ndefinability and differential algebraic groups [3].","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"48 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.04147","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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. Our result complements the long exact sequence
in definable Galois cohomology developed in More on Galois cohomology,
definability and differential algebraic groups [3].