The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group)
| dc.creator | Colliot-Th'el`ene, Jean-Louis | |
| dc.creator | Sansuc, Jean-Jacques | |
| dc.date | 2005-07-07 | |
| dc.date.accessioned | 2026-07-07T05:21:30Z | |
| dc.date.available | 2026-07-07T05:21:30Z | |
| dc.description | This is a survey on the ancient question : Let G be a reductive group over an algebraically closed field k and let V be a vector space over k with an almost free linear action of G on V. Let k(V) denote the field of rational functions on V. Is the subfield of G-invariants of k(V) purely transcendental over k ? For G connected, this is still an open question. After a discussion of general matters (various notions of rationality, various notions of quotients, the no-name lemma), we consider several specific groups G. We then discuss the unramified Brauer group of a function field and describe the work of Saltman and of Bogomolov, leading to computations of the unramified Brauer group of fields of G-invariants. The text is a thoroughly revised version of a text distributed at the 9th latino-american school (Santiago de Chile, July 1988), various versions of which had been circulated over the years. ----- Soient k un corps alg'ebriquement clos, V un espace vectoriel sur k et G un groupe r'eductif connexe sur k agissant lin'eairement sur V. Supposons l'action g'en'eriquement libre. Le sous-corps des G-invariants du corps des fonctions rationnelles k(V) est-il transcendant pur ? Le pr'esent texte est un rapport g'en'eral sur cette vieille question, encore ouverte lorsque G est connexe. On d'ecrit en particulier des travaux de Saltman et de Bogomolov sur le groupe de Brauer non ramifi'e des corps de G-invariants. Ce texte est une version tr`es remani'ee d'un texte distribu'e `a la neuvi`eme 'ecole d''et'e latino-am'ericaine (Santiago de Chile, Juillet 1988). | |
| dc.description | 59 pages, in English ; to appear in the Proceedings of the International Congress, Mumbai, January 2004 | |
| dc.identifier | https://arxiv.org/abs/math/0507154 | |
| dc.identifier | http://arxiv.org/abs/math/0507154 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/75714 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14L30, 14E08, 14M20, 14M17, 14L30 | |
| dc.title | The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group) | |
| dc.type | text |