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Representability of cohomology of finite flat abelian group schemes

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arxiv 2107.11492 v2 pith:ZL2RG5CY submitted 2021-07-24 math.AG math.NT

Daniel Bragg , Martin Olsson This is my paper · ORCID
classification math.AGmath.NT
keywords groupabelianfiniteflatcohomologyschemesrepresentabilityscheme
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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.

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Cited by 4 Pith papers

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