  We prove various finiteness and representability results for flat cohomology
of finite flat abelian group schemes. In particular, we show that if
$f:X\rightarrow \mathrm{Spec} (k)$ is a projective scheme over a field $k$ and
$G$ is a finite flat abelian group scheme over $X$ then $R^if_*G$ is an
algebraic space for all $i$. More generally, we study the derived pushforwards
$R^if_*G$ for $f:X\rightarrow S$ 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
Dieudonn\'e 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.
