On The Relative Cohomology For Algebraic Groups

Let $G$ be an algebraic group over a field $k$, and $M$ and $N$ be $G$-modules. In 1961, Hochschild showed how one can define the cohomology groups $\text{Ext}_{G}^{i}(M,N)$. Kimura, in 1965, showed that one can generalize this to get relative cohomology for algebraic groups. The original cohomology groups play an important role in understanding the representation theory of $G$, but the role of relative cohomology is still not well understood. In this paper the author expands upon the work of Kimura to prove foundational results about the relative cohomology. The author starts by giving the definitions of relative exact sequences and relative injective modules and proves a variety of basic properties for each that will be essential to define relative cohomology and obtain a relative Grothendieck spectral sequence. In particular, the induction functor will play an important role when studying the relative injective modules. Once the necessary ground work is laid, the definition of relative cohomology is given. Finally, it is stated when there is a relative Grothendieck spectral sequence, and many consequences and examples are provided.