The existence of equivariant pure free resolutions
Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 905-926

Let A=K[x 1 ,,x m ] be a polynomial ring in m variables and let d=(d 0 <<d m ) be a strictly increasing sequence of m+1 integers. Boij and Söderberg conjectured the existence of graded A-modules M of finite length having pure free resolution of type d in the sense that for i=0,,m the i-th syzygy module of M has generators only in degree d i .

This paper provides a construction, in characteristic zero, of modules with this property that are also GL(m)-equivariant. Moreover, the construction works over rings of the form A K B where A is a polynomial ring as above and B is an exterior algebra.

Soit A=K[x 1 ,,x m ] un anneau polynomial à m variables et soit d=(d 0 <<d m ) une suite strictement croissante de m+1 nombres entiers. Boij et Söderberg ont conjecturé l’existence de A-modules gradués M de longueur finie ayant une résolution pure et libre de type d dans le sens ou pour i=0,,m les générateurs du i-ème module de syzygies de M sont uniquement de degré d i .

Cet article présente une construction, en caractéristique zéro, de modules avec cette propriété qui sont aussi GL(m)-équivariants. La construction fonctionne aussi pour les anneaux de la forme A K BA est un anneau polynomial comme ci-dessus et B est une algèbre extérieure.

DOI : https://doi.org/10.5802/aif.2632
Classification:  13D02,  13C14,  14M12,  20G05
Keywords: Pure resolution, equivariant resolution, Betti diagram, Boij-Söderberg theory, Pieri map, determinantal variety
@article{AIF_2011__61_3_905_0,
     author = {Eisenbud, David and Fl\o ystad, Gunnar and Weyman, Jerzy},
     title = {The existence of equivariant pure free resolutions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {3},
     year = {2011},
     pages = {905-926},
     doi = {10.5802/aif.2632},
     mrnumber = {2918721},
     zbl = {1239.13023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_3_905_0}
}
Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy. The existence of equivariant pure free resolutions. Annales de l'Institut Fourier, Volume 61 (2011) no. 3, pp. 905-926. doi : 10.5802/aif.2632. http://www.numdam.org/item/AIF_2011__61_3_905_0/

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