The existence of equivariant pure free resolutions

Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy

doi:10.5802/aif.2632

The existence of equivariant pure free resolutions
[Existence de résolutions pures et libres equivariantes]

Eisenbud, David ¹ ; Fløystad, Gunnar ² ; Weyman, Jerzy ³

¹ Dept of Mathematics Berkeley, CA 94720 (USA)
² Matematisk Institutt Johs. Brunsgt. 12 5008 Bergen (Norway)
³ Northeastern University Department of Mathematics 360 Huntington Avenue Boston, MA 02115 (USA)

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

Abstract (VO)
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 Zbl

DOI : 10.5802/aif.2632

Classification : 13D02, 13C14, 14M12, 20G05
Keywords: Pure resolution, equivariant resolution, Betti diagram, Boij-Söderberg theory, Pieri map, determinantal variety
Mots-clés : résolution pure, résolution équivariante, diagramme de Betti, théorie de Boij-Söderberg

Affiliations des auteurs :

Eisenbud, David ¹ ; Fløystad, Gunnar ² ; Weyman, Jerzy ³

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Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy. The existence of equivariant pure free resolutions. Annales de l'Institut Fourier, Tome 61 (2011) no. 3, pp. 905-926. doi: 10.5802/aif.2632

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