TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Long exact sequence in Lie algebra cohomology

Statement

For finite-dimensional $g$ and a short exact sequence $0 \to M^{'} \to M \to M^{''} \to 0$ of $g$ -modules, the induced CE complexes form a degreewise short exact sequence and yield the natural long exact sequence

$\dots \to H^{n} (g, M^{'}) \to H^{n} (g, M) \to H^{n} (g, M^{''}) \to H^{n + 1} (g, M^{'}) \to \dots .$

Facts & Assumptions

Given: The stated short exact coefficient sequence and finite-dimensional $g$ .

[L1]

CE cochains and cohomology are as in Lie algebra cohomology.

[L2]

A short exact sequence of cochain complexes has a natural cohomology long exact sequence (The long exact sequence in cohomology).

[L3]

The coefficient maps are intertwiners (Subrepresentations, quotient representations, and intertwiners).

Proof

technique · degreewise exactness followed by the abstract theorem

1.1

For each $n$ , the space $Λ^{n} g$ is finite-dimensional. Applying ${Hom}_{k} (Λ^{n} g, -)$ preserves injections and kernels. It also preserves the given surjection: lift the images of a finite basis and extend linearly. Thus the three CE cochain spaces form a short exact sequence in every degree, including $n < 0$ , where all are zero.

L1algebra

1.2

Because the coefficient maps intertwine the action [L3], applying one before or after each of the two sums in the CE differential gives the same result. Hence the degreewise maps are cochain maps.

L1L3

2.1

Apply [L2] to the short exact sequence from steps 1.1–1.2. This gives the displayed natural sequence with the connecting map raising degree by one. At $n = 0$ it begins with the invariant subspaces; at degrees above $\dim g$ all terms vanish. The finite basis lift is a single finite construction and uses no choice principle.

L2step 1.11.2∎

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Sources

Weibel, Lie Algebra Homology and Cohomology, §§7.7–7.8 (standard reference, not scraped)