# algebraic geometry – Associated fiber bundle

Let $$k$$ be an algebraically closed field, $$X$$ a quasiprojective $$k$$ scheme, $$G$$ a smooth linear algebraic group. We consider $$f:P to X$$ to be a principal $$G$$-bundle (here I’m assuming $$f$$ to be locally trivial in the etale topology). Given now another quasi projective $$k$$ scheme $$F$$ with a $$G$$ action, we consider the (right) $$G$$-action on $$P times_k F$$ defined by: $$g cdot (p,f)=(pg,g^{-1}f) .$$

I’ve seen claimed that the quotient $$P times_k F/G$$ always exists (with no condition on $$G$$), but I was not able to prove that. In particular, I was not able to specialize the standard topological proof to the algebrogeometric context, given that we have just etale local triviality.