The existence of equivariant pure free resolutions

Eisenbud, David; Fløystad, Gunnar; Weyman, Jerzy

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

Eisenbud, David ; Fløystad, Gunnar ; Weyman, Jerzy

Annales de l'Institut Fourier, Tome 61 (2011), p. 905-926 / Harvested from Numdam

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Résumé
Abstract

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.

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.

Publié le : 2011-01-01

MR 2918721 | Zbl 1239.13023

DOI : https://doi.org/10.5802/aif.2632
Classification: 13D02, 13C14, 14M12, 20G05
Mots clés: résolution pure, résolution équivariante, diagramme de Betti, théorie de Boij-Söderberg

@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},
     volume = {61},
     year = {2011},
     pages = {905-926},
     doi = {10.5802/aif.2632},
     zbl = {1239.13023},
     mrnumber = {2918721},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 61 (2011) pp. 905-926. doi : 10.5802/aif.2632. http://gdmltest.u-ga.fr/item/AIF_2011__61_3_905_0/

Bibliographie

[1] Berele, A.; Regev, A. Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Adv. in Math., Tome 64 (1987), pp. 118-175 | Article | MR 884183 | Zbl 0617.17002

[2] Boij, M.; Söderberg, J. Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, Journal of the London Mathematical Society (2), Tome 79 (2008) no. 1, pp. 85-106 | Article | MR 2427053 | Zbl 1189.13008

[3] Buchsbaum, D. A.; Eisenbud, D. Generic free resolutions and a family of generically perfect ideals, Advances in Math., Tome 18 (1975) no. 3, pp. 245-301 | Article | MR 396528 | Zbl 0336.13007

[4] Buchsbaum, D. A.; Eisenbud, D. Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$ , Amer. J. Math., Tome 99 (1977) no. 3, pp. 447-485 | Article | MR 453723 | Zbl 0373.13006

[5] Demazure, M. A very simple proof of Bott’s theorem, Inventiones Mathematicae, Tome 34 (1976), p. 271-272 | Article | MR 414569 | Zbl 0383.14017

[6] Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry, Springer, Graduate Texts in Mathematics (1995) | MR 1322960 | Zbl 0819.13001

[7] Eisenbud, D.; Schreyer, F. Betti Numbers of Graded Modules and Cohomology of Vector Bundles, Journal of the American Mathematical Society, Tome 22 (2009) no. 3, pp. 859-888 | Article | MR 2505303 | Zbl 1213.13032

[8] Eisenbud, D.; Weyman, J. Fitting’s Lemma for $Z / 2$ -graded modules, Trans. Am. Math. Soc., Tome 355 (2003), pp. 4451-4473 | Article | MR 1990758 | Zbl 1068.13001

[9] Fløystad, G. Exterior algebra resolutions arising from homogeneous bundles, Math. Scand., Tome 94 (2004) no. 2, pp. 191-201 | MR 2053739 | Zbl 1062.14023

[10] Fløystad, G. The linear space of Betti diagrams of multigraded artinian modules, Mathematical Research Letters, Tome 17 (2010) no. 5, pp. 943-958 | MR 2727620 | Zbl 1225.13017

[11] Fulton, W.; Harris, J. Representation Theory; a first course, Springer-Verlag, Graduate Texts in Mathematics 129 (1991) | MR 1153249 | Zbl 0744.22001

[12] Herzog, J.; Kühl, M. On the Betti numbers of finite pure and linear resolutions, Comm. Algebra, Tome 12 (1984) no. 13-14, pp. 1627-1646 | Article | MR 743307 | Zbl 0543.13008

[13] Kirby, D. A sequence of complexes associated with a matrix, J. London Math. Soc. (2), Tome 7 (1974), pp. 523-530 | Article | MR 337939 | Zbl 0274.18018

[14] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford University Press, New York, Oxford Mathematical Monographs (1995) (Second edition. With contributions by A. Zelevinsky) | MR 1354144 | Zbl 0487.20007

[15] Peskine, C.; Szpiro, L. Dimension projective finie et cohomologie locale. Applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Inst. des Hautes Études Sci. Publ. Math. (1973) no. 42, pp. 47-119 | Article | Numdam | MR 374130 | Zbl 0268.13008

[16] Sam, S.; Weyman, J. Pieri Resolutions for Classical Groups (arXiv:0907.4505) | Zbl 1245.20060

[17] Weyman, J. Cohomology of vector bundles and syzygies, Cambridge University Press, Cambridge (2003) | MR 1988690 | Zbl 1075.13007