Why do twists of an algebraic group over k correspond to k-torsors over G

2012-03-17T19:28:55Z

Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$.

I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$.

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.

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 $k$?

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.

Why do twists of an algebraic group over k correspond to k-torsors over G - MathOverflow

Why do twists of an algebraic group over k correspond to k-torsors over G