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:
Published online:
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}, 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 TI - Generic free resolutions and root systems JO - Annales de l'Institut Fourier PY - 2018 DA - 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/ UR - https://doi.org/10.5802/aif.3188 DO - 10.5802/aif.3188 LA - en ID - AIF_2018__68_3_1241_0 ER -
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/
[1] The existence of generic free resolutions and related objects, Math. Scand., Tome 55 (1984), pp. 33-46 | Article | Zbl: 0557.13011
[2] What makes a complex exact, J. Algebra, Tome 25 (1973), pp. 259-268 | Article | Zbl: 0264.13007
[3] Some structure theorems for finite free resolutions, Adv. Math., Tome 1 (1974), pp. 84-139 | Article | Zbl: 0297.13014
[4] Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3, Am. J. Math., Tome 99 (1977) no. 3, pp. 447-485 | Article | Zbl: 0373.13006
[5] On the variety of complexes, Adv. Math., Tome 41 (1981), pp. 45-77 | Article | Zbl: 0471.14026
[6] Singularités rationnelles et déformations, Invent. Math., Tome 47 (1977), pp. 139-147 | Article | Zbl: 0363.14002
[7] Lie algebra homology and the Macdonald-Kac formulas, Invent. Math., Tome 34 (1976), pp. 37-76 | Article | Zbl: 0358.17015
[8] Algebraic homogeneous spaces and invariant theory, Lecture Notes in Math., Tome 1673, Springer, Berlin, 1997, vi+148 pages | Zbl: 0886.14020
[9] Topics in the homological theory of modules over commutative rings, CBMS Regional Conference Series in Mathematics, Tome 24, American Mathematical Society, 1975 | Zbl: 0302.13003
[10] Infinite Dimensional Lie algebras, Cambridge University Press, 1993, xxi+400 pages | Zbl: 0716.17022
[11] The Grothendieck-Cousin complex of an induced representation, Adv. Math., Tome 29 (1978), pp. 310-396 | Article | Zbl: 0393.20027
[12] Kac-Moody groups, their flag varieties and Representation Theory, Progress in Mathematics, Tome 204, Birkhäuser, Boston, 2002 | Zbl: 1026.17030
[13] Kostant’s formula for Kac-Moody Lie algebras, J. Algebra, Tome 149 (1992) no. 1, pp. 155-178 | Article | Zbl: 0779.17024
[14] Finite free resolutions, Cambridge Tracts in Mathematics, Tome 71, Cambridge University Press, Cambridge, UK, 1976 | Zbl: 0328.13010
[15] On the geometry of spherical varieties, Transform. Groups, Tome 19 (2014), pp. 171-223 | Article | Zbl: 1309.14001
[16] On the generic free resolutions, J. Algebra, Tome 128 (1990) no. 1, pp. 1-44 | Article | Zbl: 0688.13003
[17] Universal Complexes and the Generic Structure of Free Resolutions, Mich. Math. J., Tome 49 (2001), pp. 65-96 | Article | Zbl: 1104.13302
[18] On the structure of resolutions of length 3, J. Algebra, Tome 126 (1989) no. 1, pp. 1-33 | Article | Zbl: 0705.13008
Cited by Sources: