The higher direct images of locally constant group schemes from the Kummer log flat topology to the classical flat topology

TL;DR

This paper characterizes the higher direct images of certain group schemes under the Kummer log flat topology, revealing vanishing and isomorphism patterns, with explicit computations for specific base schemes.

Contribution

It provides a complete description of higher direct images of group schemes from the Kummer log flat site to the classical flat site, including new vanishing and isomorphism results.

Findings

Higher direct images vanish for vector space group schemes.

First higher direct image vanishes for free abelian group schemes.

Higher direct images relate to tensor products with $Z$ and $Q/Z$.

Abstract

Let $S$ be an fs log scheme, and let $F$ be a group scheme over the underlying scheme which is \'etale locally representable by (1) a finite dimensional $\mathbb{Q}$-vector space, or (2) a finite rank free abelian group, or (3) a finite abelian group. We give a full description of all the higher direct images of $F$ from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of $F$ vanish; and in case (2) the first higher direct image of $F$ vanishes and the $n$-th ($n>1$) higher direct image of $F$ is isomorphic to the $(n-1)$-th higher direct image of $F\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}$. In the end, we make some computations when the base is a standard log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.