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On The Relative Cohomology For Algebraic Groups

Gabriel T. Loos

TL;DR

This work develops a comprehensive, functorial theory of relative cohomology for algebraic groups. By defining $(G,H)$-exact sequences and $(G,H)$-injective modules and leveraging induction, the paper builds relative right derived functors $R_{(G,H)}^i$ and a relative Ext framework, culminating in a relative Grothendieck spectral sequence that generalizes classical results to the relative setting. It then applies the theory to factor groups, parabolic and Levi subgroups, and CPS coupled parabolic systems, deriving numerous isomorphisms and vanishing results under broad hypotheses and arbitrary characteristic. The results provide a flexible toolkit for computing and comparing relative cohomology across subgroups and quotients, with wide-ranging implications for representation theory and related geometric structures.

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.

On The Relative Cohomology For Algebraic Groups

TL;DR

This work develops a comprehensive, functorial theory of relative cohomology for algebraic groups. By defining -exact sequences and -injective modules and leveraging induction, the paper builds relative right derived functors and a relative Ext framework, culminating in a relative Grothendieck spectral sequence that generalizes classical results to the relative setting. It then applies the theory to factor groups, parabolic and Levi subgroups, and CPS coupled parabolic systems, deriving numerous isomorphisms and vanishing results under broad hypotheses and arbitrary characteristic. The results provide a flexible toolkit for computing and comparing relative cohomology across subgroups and quotients, with wide-ranging implications for representation theory and related geometric structures.

Abstract

Let be an algebraic group over a field , and and be -modules. In 1961, Hochschild showed how one can define the cohomology groups . 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 , 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.
Paper Structure (19 sections, 65 theorems, 140 equations)

This paper contains 19 sections, 65 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Definitions and Basic Properties
  4. Relative Exact Sequences
  5. Relative Injective Modules
  6. Direct Sums and Summands of Relative Injective Modules
  7. Induction
  8. Induction of Relative Injective Modules
  9. Relative Injective Resolutions
  10. Induction of Relative Exact Sequences
  11. Relative Right Derived Functor
  12. Relative Acyclic Modules
  13. Relative Ext
  14. Cohomology
  15. A Relative Grothendieck Spectral Sequence
  16. ...and 4 more sections

Key Result

Lemma 2.1.1

Let $H$ and $H'$ be closed subgroups of $G$ such that $H\leq H'\leq G$, if \begin{tikzcd} \cdots\arrow[r] & M_{i-1}\arrow[r, "d_{i-1}"] & M_{i}\arrow[r, "d_{i}"] & M_{i+1}\arrow[r] & \cdots \end{tikzcd}is $(G,H')$-exact, then it is also $(G,H)$-exact.

Theorems & Definitions (122)

  • Lemma 2.1.1
  • proof
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Lemma 2.3.1
  • proof
  • Lemma 2.5.1
  • proof
  • ...and 112 more