In this paper I give an explicit construction of the generic rings 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 corresponding to the format of the resolution. The ring is Noetherian if and only if the graph corresponding to a given format is a Dynkin diagram. In such case has rational singularities so it is Cohen–Macaulay. The ring is a deformation of a commutative ring which has a structure of a multiplicity free module over a product of Kac–Moody Lie algebras corresponding to the graph and a product of two general linear Lie algebras.
Dans ce papier je présente une construction explicite des anneaux génériques 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 correspondant au format de la résolution. L’anneau est noethérien si et seulement si le graphe correspondant à un format donné est un diagramme de Dynkin. Dans ce cas a des singularitées rationnelles donc il est de Cohen–Macaulay. L’anneau est une déformation d’un anneau commutatif qui possède une structure de module sans multiplicité sur un produit d’algèbres de Kac–Moody correspondant au graphe et un produit de deux algèbres de Lie linéaires générales.
Accepted:
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Keywords: Finite free resolutions, structure theorems, Kac–Moody Lie algebras
Mot clés : Résolutions libres finies, théorèmes de structure, algèbres de Kac–Moody et Lie
Weyman, Jerzy 1
@article{AIF_2018__68_3_1241_0, author = {Weyman, Jerzy}, title = {Generic free resolutions and root systems}, journal = {Annales de l'Institut Fourier}, pages = {1241--1296}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {68}, number = {3}, year = {2018}, doi = {10.5802/aif.3188}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3188/} }
TY - JOUR AU - Weyman, Jerzy TI - Generic free resolutions and root systems JO - Annales de l'Institut Fourier PY - 2018 SP - 1241 EP - 1296 VL - 68 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3188/ DO - 10.5802/aif.3188 LA - en ID - AIF_2018__68_3_1241_0 ER -
%0 Journal Article %A Weyman, Jerzy %T Generic free resolutions and root systems %J Annales de l'Institut Fourier %D 2018 %P 1241-1296 %V 68 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3188/ %R 10.5802/aif.3188 %G en %F AIF_2018__68_3_1241_0
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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