Generic free resolutions and root systems

Weyman, Jerzy

Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1241-1296.

Dans ce papier je présente une construction explicite des anneaux génériques ${\widehat{R}}_{gen}$ pour les résolutions libres de longueur 3 sur des $ℂ$-algèbres noethériennes commutatives. L’élément clé est l’algèbre de Lie de défaut introduite dans [18]. Il s’avère que l’algèbre de défaut est une sous-algèbre parabolique d’une algèbre de Kac–Moody associée au graphe $T_{p,q,r}$ correspondant au format de la résolution. L’anneau ${\widehat{R}}_{gen}$ est noethérien si et seulement si le graphe $T_{p,q,r}$ correspondant à un format donné est un diagramme de Dynkin. Dans ce cas ${\widehat{R}}_{gen}$ a des singularitées rationnelles donc il est de Cohen–Macaulay. L’anneau ${\widehat{R}}_{gen}$ est une déformation d’un anneau commutatif ${\widehat{R}}_{spec}$ qui possède une structure de module sans multiplicité sur un produit d’algèbres de Kac–Moody correspondant au graphe $T_{p,q,r}$ et un produit de deux algèbres de Lie linéaires générales.

In this paper I give an explicit construction of the generic rings ${\widehat{R}}_{gen}$ for free resolutions of length 3 over Noetherian commutative $ℂ$-algebras. The key role is played by the defect Lie algebra introduced in [18]. The defect algebra turns out to be a parabolic subalgebra in a Kac–Moody Lie algebra associated to the graph $T_{p,q,r}$ corresponding to the format of the resolution. The ring ${\widehat{R}}_{gen}$ is Noetherian if and only if the graph $T_{p,q,r}$ corresponding to a given format is a Dynkin diagram. In such case ${\widehat{R}}_{gen}$ has rational singularities so it is Cohen–Macaulay. The ring ${\widehat{R}}_{gen}$ is a deformation of a commutative ring ${\widehat{R}}_{spec}$ which has a structure of a multiplicity free module over a product of Kac–Moody Lie algebras corresponding to the graph $T_{p,q,r}$ and a product of two general linear Lie algebras.

Classification : 13D02, 13D25, 17B67, 14M05, 14M07, 14M17, 14M27
Keywords: Finite free resolutions, structure theorems, Kac–Moody Lie algebras
Mot clés : Résolutions libres finies, théorèmes de structure, algèbres de Kac–Moody et Lie

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