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

Part of: (Co)homology theory

Published online by Cambridge University Press:  04 August 2025

HEER ZHAO
HEER ZHAO*
Affiliation:
Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, P. R. China. e-mail: heer.zhao@gmail.com
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Abstract

Let S be a fine and saturated (fs) log scheme, and let F be a group scheme over the underlying scheme of S 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 nth ($n\gt 1$) 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 henselian log trait or a Dedekind scheme endowed with the log structure associated to a finite set of closed points.

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 27
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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