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Let $X$ be an affine scheme over $S$ and let $G$ be a finite group acting freely on $X$.

  1. I saw two definitions in the literature regarding "free action", the first that the map $G\times_S X\to X\times_S X$ mapping $(g,x)$ to $(gx,x)$ is a monomorphism the second that it is a closed embedding. Which one is standard?

  2. I am looking for a citable source giving that the quotient map $X\to X/G$ is finite. I saw it mentioned in some places but could not find a source.

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    $\begingroup$ In 1, the two conditions are equivalent (even when $X \to S$ is only assumed separated). Indeed, the map you wrote down is a map of schemes over $X$ (via the second projection), where $G \times_S X \to X$ is finite, hence proper. If $Z$ is the scheme-theoretic image, then $G \times_S X \to Z$ is proper [Tag 01W6], hence a closed immersion if it is a monomorphism [Tag 04XV]. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 8 at 16:57
  • $\begingroup$ @R.vanDobbendeBruyn do you want to give that as an answer so it can be accepted? You might mention for the second that $X \rightarrow X/G$ is an $G$-bundle, hence etale-locally of the form $Y \times G \rightarrow Y$, which is finite; and finiteness of morphisms descends under etale maps, thereby answring both questions. $\endgroup$
    – Ravi Vakil
    Commented Oct 10 at 18:51
  • $\begingroup$ @RaviVakil thanks, but I think mine doesn't fully answer it, so I left a comment. I have the feeling the OP is specifically looking for a citable source, which I'm not sure I can provide. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Oct 10 at 20:39
  • $\begingroup$ In the meantime I found Thm 4.16 in math.ru.nl/~bmoonen/BookAV/Quotients.pdf It is not from a paper or a book, but might be good enough $\endgroup$
    – user4231
    Commented Oct 12 at 7:10
  • $\begingroup$ @R.vanDobbendeBruyn how is it not citable? $\endgroup$
    – Ravi Vakil
    Commented Oct 29 at 0:19

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