Geodesic
The adic tame site
Documenta mathematica, Tome 26 (2021), pp. 873-945.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For every adic space $\mathscr{X}$ we construct a site $\mathscr{X}_t$, the tame site of $\mathscr{X}$. For a scheme $X$ over a base scheme $S$ we obtain a tame site by associating with $X/S$ an adic space $\mathrm{Spa}(X,S)$ and considering the tame site $\mathrm{Spa}(X,S)_t$. We examine the connection of the cohomology of the tame site with étale cohomology and compare its fundamental group with the conventional tame fundamental group. Finally, assuming resolution of singularities, for a regular scheme $X$ over a base scheme $S$ of characteristic $p > 0$ we prove a cohomological purity theorem for the constant sheaf $\mathbb{Z}/p\mathbb{Z}$ on $\mathrm{Spa}(X,S)_t$. As a corollary we obtain homotopy invariance for the tame cohomology groups of $\mathrm{Spa}(X,S)$.
Classification : 14F20, 14F35, 14G17, 14G22
Keywords: tame ramification, positive characteristic, Grothendieck topology 
@article{DOCMA_2021__26__a32,
     author = {H\"ubner, Katharina},
     title = {The adic tame site},
     journal = {Documenta mathematica},
     pages = {873--945},
     publisher = {mathdoc},
     volume = {26},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a32/}
}
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Hübner, Katharina. The adic tame site. Documenta mathematica, Tome 26 (2021), pp. 873-945. http://geodesic.mathdoc.fr/item/DOCMA_2021__26__a32/