Infinite root stacks and quasi-coherent sheaves on logarithmic schemes

TL;DR

This paper introduces infinite root stacks for logarithmic schemes, demonstrating they encode the logarithmic structure and connect to the Kummer-flat topos, extending the relationship between parabolic and quasi-coherent sheaves.

Contribution

It defines infinite root stacks for logarithmic schemes and shows they determine the structure and generalize existing correspondences with sheaves.

Findings

Infinite root stacks determine the logarithmic structure.

They recover the Kummer-flat topos of the scheme.

Extension of the parabolic sheaves and quasi-coherent sheaves correspondence.

Abstract

We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.