no. 3
Generic free resolutions and root systems
Weyman, Jerzy1
1 Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
Annales de l'Institut Fourier, Volume 68 (2018) no. 3, pp. 1241-1296.

In this paper I give an explicit construction of the generic rings 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 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 R ^ gen has rational singularities so it is Cohen–Macaulay. The ring R ^ gen is a deformation of a commutative ring 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.

Dans ce papier je présente une construction explicite des anneaux génériques 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 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 R ^ gen a des singularitées rationnelles donc il est de Cohen–Macaulay. L’anneau R ^ gen est une déformation d’un anneau commutatif 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.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3188
Classification: 13D02, 13D25, 17B67, 14M05, 14M07, 14M17, 14M27
Keywords: Finite free resolutions, structure theorems, Kac–Moody Lie algebras
Weyman, Jerzy 1

1 Department of Mathematics University of Connecticut Storrs, CT 06269 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Weyman, Jerzy. Generic free resolutions and root systems. Annales de l'Institut Fourier, Volume 68 (2018) no. 3, pp. 1241-1296. doi : 10.5802/aif.3188. https://aif.centre-mersenne.org/articles/10.5802/aif.3188/

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