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Projective dimension and commuting variety of a reductive Lie algebra

Jean-Yves Charbonnel

TL;DR

This work proves that the commuting scheme of a reductive Lie algebra ${\mathfrak g}$ is normal and Cohen–Macaulay by showing the defining ideal ${I_{\mathfrak g}}$ is prime. Central to the argument is Property (P) for simple Lie algebras, which asserts exactness of a family of cohomology complexes built from the characteristic module ${\mathrm B}_{\mathfrak g}$ and its orthogonal complement. The proof uses an induction on rank, parabolic restriction, and a detailed analysis of a filtration of complexes to transfer vanishing properties from Levi factors to ${\mathfrak g}$, culminating in depth computations via Auslander–Buchsbaum and standard cohomological criteria. The results resolve long-standing questions about the reducedness and singularities of the commuting scheme and provide a robust homological framework for studying its structure.

Abstract

The commuting variety of a reductive Lie algebra $\mathfrak{g}$ is the underlying variety of a well defined subscheme of $\mathfrak{g}\times\mathfrak{g}$. In this note, it is proved that this scheme is normal and Cohen-Macaulay. In particular, its ideal of definition is a prime ideal. As a matter of fact, this theorem results from a so called Property (P) for a simple Lie algebra. This property says that some cohomology complexes are exact.

Projective dimension and commuting variety of a reductive Lie algebra

TL;DR

This work proves that the commuting scheme of a reductive Lie algebra is normal and Cohen–Macaulay by showing the defining ideal is prime. Central to the argument is Property (P) for simple Lie algebras, which asserts exactness of a family of cohomology complexes built from the characteristic module and its orthogonal complement. The proof uses an induction on rank, parabolic restriction, and a detailed analysis of a filtration of complexes to transfer vanishing properties from Levi factors to , culminating in depth computations via Auslander–Buchsbaum and standard cohomological criteria. The results resolve long-standing questions about the reducedness and singularities of the commuting scheme and provide a robust homological framework for studying its structure.

Abstract

The commuting variety of a reductive Lie algebra is the underlying variety of a well defined subscheme of . In this note, it is proved that this scheme is normal and Cohen-Macaulay. In particular, its ideal of definition is a prime ideal. As a matter of fact, this theorem results from a so called Property (P) for a simple Lie algebra. This property says that some cohomology complexes are exact.

Paper Structure

This paper contains 39 sections, 70 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Sketch of proofs
  4. Organization of the note.
  5. General remarks and notations
  6. Characteristic module
  7. First properties of .
  8. Orthogonal complement to ${\mathrm B}_{{\mathfrak {g}_{}}}$.
  9. Restriction to a parabolic subalgebra.
  10. Proof of the main theorem
  11. Some complexes
  12. General facts
  13. Some equivalence
  14. Examples of complexes
  15. Other examples of complexes
  16. ...and 24 more sections

Key Result

Theorem 1.1

The subscheme of ${\mathfrak g}_{}\times {\mathfrak g}_{}$ defined by $I_{{\mathfrak g}}$ is Cohen-Macaulay and normal. Furthermore, $I_{{\mathfrak g}}$ is a prime ideal of ${\rm S}({\mathfrak g}_{}\times {\mathfrak g}_{})$.

Theorems & Definitions (143)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Proposition 3.1
  • ...and 133 more