On The Relative Cohomology For Algebraic Groups

TL;DR

This paper develops foundational aspects of relative cohomology for algebraic groups, extending Kimura's work, and introduces tools like relative injective modules and spectral sequences to deepen understanding of the theory.

Contribution

The paper formalizes the theory of relative cohomology for algebraic groups, including definitions, properties, and the construction of a relative Grothendieck spectral sequence.

Findings

Established basic properties of relative injective modules

Defined relative cohomology and spectral sequences

Provided examples illustrating the theory

Abstract

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.