Appears in collection : Conference on Arithmetic Geometry in honor of Luc Illusie

Let $f:X\rightarrow S$ be a proper morphism of schemes over a field $k$ of positive characteristic, and let $G$ be a finite flat abelian group scheme over $X$. In this talk I will discuss recent representability results for the derived pushforwards $R^if_*G$. Key ingredients in proving our results is the development of a theory of compactly supported flat cohomology and description in terms of the cotangent complex in some cases. This is joint work with Daniel Bragg.