ag.algebraic geometry – When is a twisted form coming from a torsor trivial?


Consider a sheaf of groups $G$, equipped with a left torsor $P$ and another left action $G$ on some $X$. Form the contracted product $P times^G X := (P times X)/sim$ where $sim$ is the antidiagonal quotient: $(g.p, x)sim (p, g.x)$.

Q1: When is $Ptimes^G X$ trivial? I.e., when do we have an isomorphism $P times^G X simeq X$?

Partial answer: $P times^G X simeq X$ over $(X/G)$ iff $P times (X/G)$ is a trivial torsor over the stack quotient $(X/G)$.

Proof: We can rewrite $P times^G X$ as a contracted product of two torsors $(P times (X/G))times^G_{(X/G)} X$. Then we contract with “$X^{-1}$” — the inverse to contracting with $X$ as a torsor over $(X/G)$ and we win. (as in B. Poonen’s Rational Points on Varieties, section 5.12.5.3)

Am I allowed to do this? This argument probably shouldn’t have to appeal to algebraic stacks and may be somewhat dubious.

Q2: If I have one isomorphism $P times^G X simeq X$, can I choose another one that lies over $(X/G)$? Or at least is $G$-equivariant?

Q3: Is there a natural way to write the triviality of such a twisted form?

I first thought $P times^G X simeq X$ iff $P$ was trivial, which is clearly false for trivial actions on $X$. Then I was excited to have the pullback $* to BG$ represent triviality of the twisted form $P times^G X$ as well as the torsor $P$. Is there a natural representative of the sheaf of isomorphisms between $P times^G X$ and $X$?

These can all be sheaves, although I’m primarily interested in $G = GL_n, PGL_n, SL_n$, etc. acting on $X = mathbb{A}^n, mathbb{P}^n$ as appropriate. More ambitious is $G = text{Aut}(X)$ for even simple $X$. I’d be happy with answers in any level of generality.

Due Diligence Statement: I’m a novice in the area of “twisted forms” of varieties, so I apologize if the above is evident or obtuse. I checked all the “similar questions” listed here and couldn’t find an answer.