{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T08:27:06Z","timestamp":1772267226681,"version":"3.50.1"},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6128,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1997,6]]},"abstract":"<jats:p>In this paper we develop some basic features of Galois cohomology, specifically the connection between first Galois cohomology groups and principal homogeneous spaces, in a model-theoretic context. \u201cDescent theory\u201d also fits into our approach.<\/jats:p><jats:p>The model theory involved is elementary, and the reader is referred to [2]. It should be said that we make crucial use of <jats:italic>M<jats:sup>eq<\/jats:sup><\/jats:italic> in our analysis. The reader is also referred to Poizat's seminal paper \u201cUne theorie de Galois imaginaire\u201d ([6]). Although our results do not depend on Poizat's work, it is in his paper that the model-theoretic context is suggested for a generalised treatment of Galois theory.<\/jats:p><jats:p>Nothing in this paper is particularly deep. We are concerned mainly with translating between the Galois cohomological language and the language of definable sets and definable families of definable sets. We will introduce (in a suitable context) the notion of a definable cocycle (from an automorphism group <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016315_inline1\"\/> to a definable group <jats:italic>G<\/jats:italic>). The (classical) situation of profinite <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200016315_inline1\"\/> and continuous cocycles will be a special case. Kolchin's theory of constrained cohomology will be another special case, and our results yield a substantially simpler proof of his Theorem 5 from Chapter VII of [4]. In any case model-theorists will see that definable cocycles correspond to objects with which they are already quite familiar\u2014commuting families of definable bijections.<\/jats:p>","DOI":"10.2307\/2275542","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T19:01:20Z","timestamp":1146942080000},"page":"487-492","source":"Crossref","is-referenced-by-count":10,"title":["Remarks on Galois cohomology and definability"],"prefix":"10.1017","volume":"62","author":[{"given":"Anand","family":"Pillay","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200016315_ref004","volume-title":"Differential algebraic groups","author":"Kolchin","year":"1985"},{"key":"S0022481200016315_ref006","first-page":"1151","volume":"48","author":"Poizat","year":"1983","journal-title":"Une theorie de Galois imaginaire"},{"key":"S0022481200016315_ref003","volume-title":"Differential algebra and algebraic groups","author":"Kolchin","year":"1973"},{"key":"S0022481200016315_ref005","volume-title":"Pacific Journal of Math","author":"Pillay"},{"key":"S0022481200016315_ref001","first-page":"40","article-title":"Methodes Galoisiennes et courbes de genre 1","volume":"89","author":"Chatelet","year":"1946","journal-title":"Ann. Univ. Lyon Sect. A"},{"key":"S0022481200016315_ref002","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200016315_ref007","volume-title":"Lecture Notes","volume":"5","author":"Serre","year":"1994"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200016315","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T16:55:54Z","timestamp":1557593754000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200016315\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,6]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1997,6]]}},"alternative-id":["S0022481200016315"],"URL":"https:\/\/doi.org\/10.2307\/2275542","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,6]]}}}