Stabilisation of the LHS spectral sequence for algebraic groups

TL;DR

This paper investigates the stabilization of the Lyndon--Hochschild--Serre spectral sequence for algebraic groups, proposing a conjecture that the spectral sequence collapses at the E2 page under certain conditions, simplifying extension computations.

Contribution

It introduces a conjecture that the spectral sequence stabilizes at E2 for algebraic groups and provides conditions under which low-dimensional terms remain unchanged at the limit.

Findings

Conjecture that E2=E_infinity for the spectral sequence.

Conditions identified for low-dimensional terms to stabilize.

Discussion on implications for extension calculations in algebraic groups.

Abstract

In this note, we consider the Lyndon--Hochschild--Serre spectral sequence corresponding to the first Frobenius kernel of an algebraic group G, computing the extensions between simple $G$-modules. We state and discuss a conjecture that $E_2=E_\infty$ and provide general conditions for low-dimensional terms on the $E_2$-page to be the same as the corresponding terms on the $E_{\infty}$-page, i.e. its abutment.