Let $X, Y$ be two quasi-affine schemes over $\mathbb{C}$. Let $G$ be a reductive algebraic group. Suppose that we are given an action of $G$ on $X,Y$ and a $G$-equivariant finite morphism $X \rightarrow Y$. Is it true that the induced morphism between algebraic stacks $X/G \rightarrow Y/G$ is still finite? What about the case when $G$ is a torus? Thanks!
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$\begingroup$ What is the pull-back of the $G$-torsor $Y\to Y/G$ along $X/G\to Y/G$? (Hint: It is a $G$-torsor over $X/G$ with a $G$-equivariant map to $Y$...) Now, if $X\to S$ is a finitely presented morphism of algebraic stacks and $T\to S$ is fppf, then $X\to S$ is finite IFF $X_T\to T$ is finite. Or am I wrong? $\endgroup$– Ariyan JavanpeykarCommented May 16, 2018 at 21:08
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