A note on short and long exact sequences in the BBG construction of complexes from complexes

TL;DR

This paper explores the cohomology of Bernstein-Gelfand-Gelfand sequences relevant to PDE analysis, using exact sequences and spectral sequences to extend and interpret these results.

Contribution

It introduces a method to derive cohomology of BGG sequences via long exact sequences, extending to non-injective/surjective cases with spectral sequence interpretation.

Findings

Cohomology of BGG sequences can be obtained through long exact sequences.

Extension to non-injective/surjective cases using short exact sequences.

Spectral sequences provide an interpretative framework.

Abstract

We first show how the cohomology of some Bernstein-Gelfand-Gelfand (BGG) sequences that are important for the numerical analysis of partial differential equations, can be obtained through the construction of a long exact sequence connecting cohomology groups. Then we explain the extension of this result to the non-injective/surjective case through the systematic use of short exact sequences of complexes and their associated long exact sequences of cohomology groups. Finally an interpretation in terms of spectral sequences is given.