Representability of cohomology of finite flat abelian group schemes

2021/07/24 by Daniel Bragg, Martin Olsson, Bragg, Daniel +1 · 6 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.11492

openalex publication_date 2021/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if f\colon X→ Spec(k) is a projective scheme over a field k and G is a finite flat abelian group scheme over X then Rⁿf_*G is representable for all n. More generally, we study the derived pushforwards Rⁿf_*G for f\colon X→ S a projective morphism and G a finite flat abelian group scheme over X. We also define compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height 1, and prove higher categorical versions of our main representability results.

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