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The structure of the Boij-Söderberg posets

Author: David Cook II
Journal: Proc. Amer. Math. Soc. 139 (2011), 2009-2015
MSC (2010): Primary 05E45, 06B23, 13C14
Published electronically: December 1, 2010
MathSciNet review: 2775378
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Abstract: Boij and Söderberg made a pair of conjectures, which were subsequently proven by Eisenbud and Schreyer and then extended by Boij and Söderberg, concerning the structure of Betti diagrams of graded modules. In the theory, a particular family of posets and their associated order complexes play an integral role. We explore the structure of this family. In particular, we show that the posets are bounded complete lattices and the order complexes are vertex-decomposable, hence Cohen-Macaulay and squarefree glicci.

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  • 1. Anders Björner and Michelle Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), no. 1, 323-341. MR 690055 (84f:06004)
  • 2. Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945-3975. MR 1401765 (98b:06008)
  • 3. Mats Boij and Jonas Söderberg, Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen-Macaulay case, arXiv e-prints (2008).
  • 4. Mats Boij and Jonas Söderberg, Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture, J. Lond. Math. Soc. (2) 78 (2008), no. 1, 85-106. MR 2427053 (2009g:13018)
  • 5. David Cook II, Simplicial decomposability, The Journal of Software for Algebra and Geometry 2 (2010), 20-23.
  • 6. P. Di Francesco, $ {\rm SU}(N)$ meander determinants, J. Math. Phys. 38 (1997), no. 11, 5905-5943. MR 1480837 (99k:05019)
  • 7. David Eisenbud and Frank-Olaf Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), no. 3, 859-888. MR 2505303
  • 8. Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at
  • 9. Uwe Nagel and Tim Römer, Glicci simplicial complexes, J. Pure Appl. Algebra 212 (2008), no. 10, 2250-2258. MR 2426505 (2009c:13025)
  • 10. J. Scott Provan and Louis J. Billera, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Oper. Res. 5 (1980), no. 4, 576-594. MR 593648 (82c:52010)

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Additional Information

David Cook II
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027

Keywords: Boij-Söderberg theory, lattice, order complex, vertex-decomposable
Received by editor(s): June 10, 2010
Published electronically: December 1, 2010
Additional Notes: Part of the work for this paper was done while the author was partially supported by the National Security Agency under grant number H98230-09-1-0032.
Communicated by: Irena Peeva
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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