Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Heer Zhao

TL;DR

This work provides a comprehensive description of the higher direct images $R^i\varepsilon_{\mathrm{fl}*}F$ when passing from the Kummer log flat topology to the classical flat topology on locally finite fs log schemes, for $F$ étale-locally modeled by a finite-dimensional ${\mathbb Q}$-vector space, a finite rank free abelian group, or a finite abelian group. It delivers explicit formulas and vanishing results: in the torsion case with $l$-power torsion, a sharp formula in terms of $j_{\mathrm{fl}!}$ and $\bigwedge^i(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}})$; in the rational-vector-space case, full vanishing; and in the finite-rank torsion-free case, a decomposition of higher direct images into sums over primes $l$ involving $F\otimes{\mathbb Q}_l/{\mathbb Z}_l$ and logarithmic exterior powers. The authors develop a robust toolkit using Čech cohomology, Leray spectral sequences, and reductions to strictly henselian local settings, and they apply the results to standard log traits and Dedekind domains to obtain concrete computations and comparisons with étale cohomology. The results have implications for log abelian varieties with constant degeneration and provide a structured way to understand the interplay between log-flat and classical flat cohomology in arithmetic geometry. Overall, the paper advances the understanding of cohomology on fs log schemes and offers practical formulas for computations and applications in log geometry and arithmetic geometry.

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.

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

TL;DR

This work provides a comprehensive description of the higher direct images when passing from the Kummer log flat topology to the classical flat topology on locally finite fs log schemes, for étale-locally modeled by a finite-dimensional -vector space, a finite rank free abelian group, or a finite abelian group. It delivers explicit formulas and vanishing results: in the torsion case with -power torsion, a sharp formula in terms of and ; in the rational-vector-space case, full vanishing; and in the finite-rank torsion-free case, a decomposition of higher direct images into sums over primes involving and logarithmic exterior powers. The authors develop a robust toolkit using Čech cohomology, Leray spectral sequences, and reductions to strictly henselian local settings, and they apply the results to standard log traits and Dedekind domains to obtain concrete computations and comparisons with étale cohomology. The results have implications for log abelian varieties with constant degeneration and provide a structured way to understand the interplay between log-flat and classical flat cohomology in arithmetic geometry. Overall, the paper advances the understanding of cohomology on fs log schemes and offers practical formulas for computations and applications in log geometry and arithmetic geometry.

Abstract

Let be an fs log scheme, and let be a group scheme over the underlying scheme which is étale locally representable by (1) a finite dimensional -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 from the Kummer log flat site to the classical flat site. In particular, we show that: in case (1) the higher direct images of vanish; and in case (2) the first higher direct image of vanishes and the -th () higher direct image of is isomorphic to the -th higher direct image of . 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.
Paper Structure (17 sections, 24 theorems, 141 equations)

This paper contains 17 sections, 24 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. The higher direct images
  3. Case of finite abelian $l$-groups
  4. Case of rational vector spaces
  5. Case of finite rank torsion-free abelian groups
  6. Examples
  7. Some general results
  8. Discrete valuation rings
  9. Global Dedekind domains
  10. Sites and sheaves
  11. Sites
  12. Continuous functors
  13. Cocontinuous functors
  14. The Kummer log flat site
  15. Morphisms of sites associated to a strict open immersion
  16. ...and 2 more sections

Key Result

Theorem 1.1

(See also Theorem 2.9) Let $S$ be a locally noetherian fs log scheme. Let $l$ be a prime number, $U$ the open locus on $S$ where $l$ is invertible, and $j:U\hookrightarrow S$ the corresponding strict open immersion. Let $F$ be a finite étale group scheme over the underlying scheme of $S$, and we end for $i\geq1$, where $j_{\mathrm{kfl}}:(\mathrm{fs}/U)_{\mathrm{kfl}}\to (\mathrm{fs}/S)_{\mathrm{kf

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 43 more