Tame distillation and desingularization by $p$-alterations

TL;DR

This paper improves Gabber's alteration theorem by removing the need for primes invertible on the scheme, enabling desingularization of schemes over quasi-excellent threefolds using only primes dividing the scheme's characteristic.

Contribution

It introduces a tame distillation theorem that decomposes alterations into tame Galois and characteristic-based alterations, strengthening desingularization methods.

Findings

Any scheme of finite type over a quasi-excellent threefold can be desingularized by a characteristic-based alteration.

The tame distillation theorem allows splitting alterations into tame Galois and characteristic-based parts.

The proof leverages a tameness theorem related to algebraic extensions of valued fields with no non-trivial p-extensions.

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 non-invertible 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 of residue characteristic $p$ has no non-trivial $p$-extensions then any its algebraic extension is tame.