Representability of cohomology of finite flat abelian group schemes

TL;DR

This paper establishes key finiteness and representability results for the cohomology of finite flat abelian group schemes, extending to derived pushforwards, compactly supported cohomology, and higher categorical contexts.

Contribution

It proves that derived pushforwards of finite flat abelian group schemes are representable over projective schemes and introduces new cohomological frameworks for these group schemes.

Findings

Derived pushforwards are representable for all degrees.

Cohomology can be described via the cotangent complex for height 1 group schemes.

Higher categorical versions of representability are established.

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.