Abstract
We strengthen Gabber's $l$'-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a $\mathrm{char}(X)$-alteration, i.e., an alteration whose order is only divisible by primes noninvertible on $X$. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of $X$ can be split into a composition of a tame Galois alteration and a $\mathrm{char}(X)$-alteration. The proof of the distillation theorem is based on the following tameness theorem that we deduce from a theorem of M. Pank: if a valued field $k$ of residue characteristic $p$ has no nontrivial $p$-extensions, then any algebraic extension $l/k$ is tame.
Citation
Michael Temkin. "Tame distillation and desingularization by $p$-alterations." Ann. of Math. (2) 186 (1) 97 - 126, July 2017. https://doi.org/10.4007/annals.2017.186.1.3
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