Generic free resolutions and root systems
Annales de l'Institut Fourier, Volume 68 (2018) no. 3, p. 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 -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 -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 : 2017-03-14
Accepted : 2017-06-15
Published online : 2018-05-04
DOI : https://doi.org/10.5802/aif.3188
Classification:  13D02,  13D25,  17B67,  14M05,  14M07,  14M17,  14M27
Keywords: Finite free resolutions, structure theorems, Kac–Moody Lie algebras
@article{AIF_2018__68_3_1241_0,
     author = {Weyman, Jerzy},
     title = {Generic free resolutions and root systems},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {3},
     year = {2018},
     pages = {1241-1296},
     doi = {10.5802/aif.3188},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_3_1241_0}
}
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/item/AIF_2018__68_3_1241_0/

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