The existence of equivariant pure free resolutions

Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy

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

Soit $A=K[x_1,\cdots ,x_m]$ un anneau polynomial à $m$ variables et soit $\mathbf{d}=(d_0<\cdots <d_m)$ 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 $\mathbf{d}$ dans le sens ou pour $i=0,\cdots ,m$ 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 $GL\left(m\right)$-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.

Let $A=K[x_1,\cdots ,x_m]$ be a polynomial ring in $m$ variables and let $\mathbf{d}=(d_0<\cdots <d_m)$ 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 $\mathbf{d}$ in the sense that for $i=0,\cdots ,m$ 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 $GL\left(m\right)$-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.

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

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