The existence of equivariant pure free resolutions

Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy

Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy

Annales de l'Institut Fourier, Volume 61 (2011) no. 3, p. 905-926

Abstract
Résumé

Let $A = K [x_{1}, \dots, x_{m}]$ be a polynomial ring in $m$ variables and let $d = (d_{0} < \dots < d_{m})$ 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 $i = 0, \dots, m$ 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 $G L (m)$ -equivariant. Moreover, the construction works over rings of the form $A \otimes_{K} B$ where $A$ is a polynomial ring as above and $B$ is an exterior algebra.

Soit $A = K [x_{1}, \dots, x_{m}]$ un anneau polynomial à $m$ variables et soit $d = (d_{0} < \dots < d_{m})$ 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 $i = 0, \dots, m$ 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 $G L (m)$ -équivariants. La construction fonctionne aussi pour les anneaux de la forme $A \otimes_{K} B$ où $A$ est un anneau polynomial comme ci-dessus et $B$ est une algèbre extérieure.

MR 2918721 | Zbl 1239.13023

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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