Representability of cohomology of finite flat abelian group schemes
Daniel Bragg, Martin Olsson
TL;DR
The paper develops a comprehensive framework for representability of flat cohomology of finite flat abelian group schemes. Central to the approach is the Kan-extension analysis of height-1 group schemes, relating flat cohomology to cotangent-data via Hoobler sequences and derived constructions, and extended through animation and pro-objects to handle general bases and formal neighborhoods. It proves global and generic representability results, establishes projective-bundle formulas in flat cohomology, and introduces a robust theory of algebraic and stably algebraic complexes to control cohomology sheaves. The work advances understanding of when Rf_*G is representable or finitely presented, with applications to finiteness over finite fields and to generic representability over base schemes, and it lays groundwork for future extensions to mixed characteristic and broader coefficient categories.
Abstract
We prove various finiteness and representability results for cohomology of finite flat abelian group schemes. In particular, we show that if $f\colon X\rightarrow \mathrm{Spec}(k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^nf_*G$ is representable for all $n$. More generally, we study the derived pushforwards $R^nf_*G$ for $f\colon X\rightarrow S$ 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.