Compactifications of Log Morphisms

TL;DR

This paper introduces a new concept of relative log schemes with boundary, extending the sheaf of differentials and enabling the development of de Rham and crystalline cohomology theories for semistable log schemes that are not necessarily proper.

Contribution

It defines relative log schemes with boundary and explores their properties, including smoothness and differential sheaves, to facilitate cohomology theories for non-proper semistable log schemes.

Findings

Extension of sheaf of differentials to compactifications

Definition of smoothness for relative log schemes with boundary

Potential application to de Rham and crystalline cohomology theories

Abstract

We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally extends to this compactification and there is a notion of smoothness for such data. We indicate how this weak sort of compactification may be used to develop useful de Rham and crystalline cohomology theories for semistable log schemes over the log point over a field which are not necessarily proper.