Logarithmic geometry beyond fs

TL;DR

This paper extends logarithmic geometry beyond finite type conditions, introducing new notions like sfp morphisms to study semistable models over general valuation rings.

Contribution

It develops foundational concepts for logarithmic structures beyond fs, including sfp morphisms and extended Kummer étale theory.

Findings

Sfp morphisms are locally approximable by finitely presented fs log scheme maps.

Extended Kummer étale site and fundamental group beyond fs cases.

Surprising finiteness of underlying scheme maps over valuation rings.

Abstract

We develop the foundations of logarithmic structures beyond the standard finiteness conditions. The motivation is the study of semistable models over general valuation rings. The key new notion is that of a morphism of finite presentation up to saturation (sfp), which is one that is qcqs and which is locally isomorphic to the saturated base change of a finitely presented morphism between fs log schemes. As in the case of schemes, sfp maps can (locally on the base) be approximated by maps between fs log schemes of finite type over $\mathbb{Z}$. Based on sfp maps, we define smooth, \'etale, and Kummer \'etale maps. Importantly, the maps of schemes underlying such maps are no longer of finite type in general, though surprisingly they are if the base is the spectrum of a valuation ring with algebraically closed field of fractions. These foundations allow us to extend beyond the fs case the theory of the Kummer \'etale site and of the Kummer \'etale fundamental group.