no. 3
Generic free resolutions and root systems
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 
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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/

[1] Bruns, Winfried The existence of generic free resolutions and related objects, Math. Scand., Tome 55 (1984), pp. 33-46 | Article | Zbl: 0557.13011

[2] Buchsbaum, David A.; Eisenbud, David What makes a complex exact, J. Algebra, Tome 25 (1973), pp. 259-268 | Article | Zbl: 0264.13007

[3] Buchsbaum, David A.; Eisenbud, David Some structure theorems for finite free resolutions, Adv. Math., Tome 1 (1974), pp. 84-139 | Article | Zbl: 0297.13014

[4] Buchsbaum, David A.; Eisenbud, David 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] De Concini, Corrado; Strickland, Elisabetta On the variety of complexes, Adv. Math., Tome 41 (1981), pp. 45-77 | Article | Zbl: 0471.14026

[6] Elkik, Renée Singularités rationnelles et déformations, Invent. Math., Tome 47 (1977), pp. 139-147 | Article | Zbl: 0363.14002

[7] Garland, Howard; Lepowsky, James Lie algebra homology and the Macdonald-Kac formulas, Invent. Math., Tome 34 (1976), pp. 37-76 | Article | Zbl: 0358.17015

[8] Grosshans, Frank D. Algebraic homogeneous spaces and invariant theory, Lecture Notes in Math., Tome 1673, Springer, Berlin, 1997, vi+148 pages | Zbl: 0886.14020

[9] Hochster, Melvin 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] Kac, Victor G. Infinite Dimensional Lie algebras, Cambridge University Press, 1993, xxi+400 pages | Zbl: 0716.17022

[11] Kempf, George The Grothendieck-Cousin complex of an induced representation, Adv. Math., Tome 29 (1978), pp. 310-396 | Article | Zbl: 0393.20027

[12] Kumar, Shrawan Kac-Moody groups, their flag varieties and Representation Theory, Progress in Mathematics, Tome 204, Birkhäuser, Boston, 2002 | Zbl: 1026.17030

[13] Liu, Lishi Kostant’s formula for Kac-Moody Lie algebras, J. Algebra, Tome 149 (1992) no. 1, pp. 155-178 | Article | Zbl: 0779.17024

[14] Northcott, D. G. Finite free resolutions, Cambridge Tracts in Mathematics, Tome 71, Cambridge University Press, Cambridge, UK, 1976 | Zbl: 0328.13010

[15] Perrin, Nicolas On the geometry of spherical varieties, Transform. Groups, Tome 19 (2014), pp. 171-223 | Article | Zbl: 1309.14001

[16] Pragacz, Piotr; Weyman, Jerzy On the generic free resolutions, J. Algebra, Tome 128 (1990) no. 1, pp. 1-44 | Article | Zbl: 0688.13003

[17] Tchernev, Alexandre B. Universal Complexes and the Generic Structure of Free Resolutions, Mich. Math. J., Tome 49 (2001), pp. 65-96 | Article | Zbl: 1104.13302

[18] Weyman, Jerzy On the structure of resolutions of length 3, J. Algebra, Tome 126 (1989) no. 1, pp. 1-33 | Article | Zbl: 0705.13008

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