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Register a new userRepresentability of cohomology of finite flat abelian group schemes
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We prove various finiteness and representability results for flat cohomology of finite flat abelian group schemes. In particular, we show that if $f:Xrightarrow mathrm{Spec} (k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^if_*G$ is an algebraic space for all $i$. More generally, we study the derived pushforwards $R^if_*G$ for $f:Xrightarrow S$ a projective morphism and $G$ a finite flat abelian group scheme over $X$. We also develop a theory of compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height $1$, and relate the Dieudonne modules of the group schemes $R^if_*mu _p$ to cohomology generalizing work of Illusie. A higher categorical version of our main representability results is also presented.
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We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of $mathbb{Q}_p$. We determine the connected components of this space and describe its irreducible components. These results prove a modified version of a conjecture of Kisin.
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