On the structure of cohomology rings of p-nilpotent Lie algebras

TL;DR

This paper explores the structure of cohomology rings of p-nilpotent Lie algebras, using spectral sequences to identify conditions for their Cohen-Macaulay property and ring isomorphisms.

Contribution

It introduces criteria for spectral sequence collapse and provides examples where cohomology rings are or are not Cohen-Macaulay, advancing understanding of their algebraic structure.

Findings

Spectral sequence collapse implies Cohen-Macaulay cohomology rings.

Conditions are identified for when the cohomology ring is isomorphic to the E2-term.

Examples show the failure of ring isomorphism and Cohen-Macaulay property in some cases.

Abstract

In this paper the authors investigate the structure the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose $E_{2}$-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen-Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as ring with the $E_{2}$-term. We present criteria for the collapsing of this spectral sequence and provide many examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.