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

Abstract

Let $S$ be an fs log scheme, and let $F$ be a group scheme over the underlying scheme which is étale 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.