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

2023/01/20 by Heer Zhao, Zhao, Heer
Mathematics · #14A21 (secondary) #14F20 14F20 (primary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2301.08566

openalex publication_date 2023/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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⊗_ℤℚ/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.

Related

• Comparison of Kummer logarithmic topologies with classical topologies II
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