# 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. Posted on Categories Articles