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I know that normality is étale local i.e., if we have $Y\rightarrow X$ étale with $X$ normal then $Y$ is normal. But is this true if we replace "normal" with "satisfies Serre's $S_2$"? More generally, does it hold for Serre's $S_k$?

Say $Y\rightarrow X$ is finite étale, then $Y$ is proper. So maybe we can use proper base change to conclude by the cohomological characterization of $S_k$ (see here).

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    $\begingroup$ The $S_n$ property can be formulated via vanishing of local cohomology. This can be checked after completion. $\endgroup$
    – Jason Starr
    Commented Jun 8 at 9:07
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    $\begingroup$ By flatness of $Y\to X$, a regular sequence in a local ring on $X$ produces a regular sequence in a local ring on $Y$. $\endgroup$
    – Piotr Achinger
    Commented Jun 8 at 13:52
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    $\begingroup$ While @PiotrAchinger's argument looks easier, this is also Tag 0352. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 8 at 15:53
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    $\begingroup$ It's even syntomic local, see Tag 036A. Also, there are different notions of "local"; the Stacks project uses this definition which in turn depends on the exact definition of coverings for each topology one considers. $\endgroup$
    – Johan
    Commented Jun 8 at 19:08

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