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.

💡 Research Summary

The paper revisits and substantially expands the theory of relative cohomology for algebraic groups, originally introduced by Kimura in the mid‑1960s, and places it on a modern, characteristic‑independent footing. After a brief historical motivation—recalling Hochschild’s Ext‑groups for rational G‑modules and Kimura’s (G,H)‑relative Ext in characteristic zero—the author sets out to develop a functorial framework that works over an arbitrary field.

Section 2 establishes the basic language. A short exact sequence of rational G‑modules is called (G,H)‑exact if each kernel splits as an H‑module direct summand; equivalently there exist H‑module homomorphisms providing a contracting homotopy. Lemmas show that (G,H′)‑exactness implies (G,H)‑exactness for H⊆H′ and that tensoring with a G‑module or applying Hom_H(E,–) preserves (G,H)‑exactness under mild hypotheses. The notion of a (G,H)‑injective module is defined analogously to ordinary injectivity, and basic closure properties (under direct sums, direct summands, and enlargement of H) are proved.

Section 3 is the technical heart. Using the adjoint pair (res_G^H, ind_G^H) the author proves that induction carries relative injectives to relative injectives: if M is (H, H∩K)‑injective then ind_G^H(M) is (G, K)‑injective (Theorem 3.2.1). The proof proceeds by restricting a (G,K)‑exact sequence to H, applying the (H, H∩K)‑injectivity of M, and then re‑inducing to obtain a G‑map that lifts a given morphism. Corollaries specialize to the case K=H, showing that ind_G^H(M) is always (G,H)‑injective, and that every rational G‑module embeds as an H‑direct summand of such an induced module. Lemma 3.3.1 gives a complete characterization: a G‑module is (G,H)‑injective iff it is a direct summand of some ind_G^H(N). This mirrors the classical description of injectives as direct summands of coinduced modules.

Section 4 introduces relative right derived functors R^i F_{G/H} for any left exact functor F: Mod(G)→Mod(G′). The author identifies sufficient conditions—chiefly that ind_G^H be exact on (G,H)‑exact sequences—to guarantee the existence of a Grothendieck spectral sequence \