Why do twists of an algebraic group over k correspond to k-torsors over G - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:14:23Z http://mathoverflow.net/feeds/question/91488 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91488/why-do-twists-of-an-algebraic-group-over-k-correspond-to-k-torsors-over-g Why do twists of an algebraic group over k correspond to k-torsors over G Harry 2012-03-17T19:28:55Z 2012-03-17T19:28:55Z <p>Let \$G\$ be an algebraic group over a field \$k\$. Let \$k^s\$ be the separable closure of \$k\$.</p> <p>I can't seem to figure out why isomorphism classes of twists of \$G\$ correspond to \$k\$-torsors over \$G\$.</p> <p>It's easy to see that a \$k\$-torsor over \$G\$ gives a twist of \$G\$, but the other way around isn't clear to me. It is bound to involve something from descent theory that I don't know.</p> <p>In fact, my problem is that I can't seem to define an action of \$G\$ on \$X\$. I can only seem to get an action of \$G_{k^s}\$ on \$X_{k^s}\$ via the isomorphism of \$X_{k^s}\$ with \$G_{k^s}\$. Why does this action descend to \$k\$?</p> <p>Let me precise that a twist of \$G\$ is a variety \$X\$ over \$k\$ such that \$X_{k^s}\$ is isomorphic to \$G_{k^s}\$ as a variety.</p>