$PGL_n$ action on an affine variety

Question

Let $X$ be an affine variety on which $PGL_n$ acts freely. Then how to see that over the quotient variety $X / PGL_{n}$, there is a bundle of central simple algebras $M_n(k) \times^{PGL_{n}} X$. I am not even able to understand the notation $M_n(k) \times^{PGL_{n}} X$. Can anyone please let me know..

Answer 1

If a group $G$ acts on two varieties $X$ and $Y$ (on the left and right respectively) then one talks of an action on $X\times Y$ defined in the following way: $g.(x,y) = (xg^{-1}, gy)$. The quotient for this action is denoted by $X\times ^G Y$.