Comparison of Kummer logarithmic topologies with classical topologies II

TL;DR

This paper investigates the properties of higher direct images of smooth commutative group schemes in Kummer log flat topology, providing explicit descriptions and conditions under which these images vanish or can be computed, especially over specific base schemes.

Contribution

It offers new explicit descriptions of higher direct images in Kummer log flat topology and identifies conditions for their vanishing, extending classical results to the logarithmic setting.

Findings

Higher direct images are torsion in the Kummer log flat site.

Explicit descriptions of second higher direct images for certain group schemes.

Vanishing of second higher direct images under rank and base scheme conditions.

Abstract

We show that the higher direct images of smooth commutative group schemes from the Kummer log flat site to the classical flat site are torsion. For (1) smooth affine commutative schemes with geometrically connected fibers, (2) finite flat group schemes, (3) extensions of abelian schemes by tori, we give explicit description of the second higher direct image. If the rank of the log structure at any geometric point of the base is at most one, we show that the second higher direct image is zero for group schemes in case (1), case (3), and certain subcase of case (2). If the underlying scheme of the base is over $\mathbb{Q}$ or of characteristic $p>0$, we can also give more explicit description of the second higher direct image of group schemes in case (1), case (3), and certain subcase of case (3). Over standard Henselian log traits with finite residue field, we compute the first and the second Kummer log flat cohomology group with coefficients in group schemes in case (1), case (3), and certain subcase of case (3).