On The Relative Cohomology For Algebraic Groups

Abstract: Let $G$ be an algebraic group over a field $k$. In 1961, Hochschild showed how one can define $\text{Ext}_{G}^{i}(M,N)$, and consequently the cohomology groups, for $G$ and $G$-modules $N$ and $M$. Afterwards, in 1965, Kimura showed that one can generalize this to get relative cohomology for algebraic groups. The cohomology groups play an important role in understanding the representation theory of $G$, but the role of relative cohomology has been left untouched. In this paper the author expands upon the work of Kimura to get more general results. In particular, we will state when there is a relative Grothendieck spectral sequence, which we will then use to show when relative cohomology is an interesting invariant.