Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-20T14:32:52.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Filtering free resolutions

Part of: Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  18 March 2013

David Eisenbud ,
Daniel Erman  and
Frank-Olaf Schreyer
David Eisenbud
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, USA email eisenbud@math.berkeley.edu
Daniel Erman
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA email erman@umich.edu
Frank-Olaf Schreyer
Affiliation:
Mathematik und Informatik, Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany email schreyer@math.uni-sb.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A recent result of Eisenbud–Schreyer and Boij–Söderberg proves that the Betti diagram of any graded module decomposes as a positive rational linear combination of pure diagrams. When does this numerical decomposition correspond to an actual filtration of the minimal free resolution? Our main result gives a sufficient condition for this to happen. We apply it to show the non-existence of free resolutions with some plausible-looking Betti diagrams and to study the semigroup of quiver representations of the simplest ‘wild’ quiver.

Keywords

syzygiesfree resolutionsBoij–Söderberg theory

MSC classification

Secondary: 13C05: Structure, classification theorems 13C14: Cohen-Macaulay modules 13D02: Syzygies, resolutions, complexes
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 5 , May 2013 , pp. 754 - 772
Copyright
© The Author(s) 2013 

References

Boij, M. and Söderberg, J., Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture, J. Lond. Math. Soc. (2) 78 (2008), 85–106; MR 2427053(2009g:13018).Google Scholar
Boij, M. and Söderberg, J., Betti numbers of graded modules and the multiplicity conjecture in the non-Cohen–Macaulay case, Algebra Number Theory 6 (2012), 437–454.Google Scholar
Buchsbaum, D. A. and Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), 447–485; MR 0453723(56 #11983).Google Scholar
Eisenbud, D., Fløystad, G. and Weyman, J., The existence of equivariant pure free resolutions, Ann. Inst. Fourier (Grenoble) 61 (2011), 905–926; MR 2918721.Google Scholar
Eisenbud, D. and Schreyer, F.-O., Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), 859–888; MR 2505303(2011a:13024).Google Scholar
Eisenbud, D. and Schreyer, F.-O., Cohomology of coherent sheaves and series of supernatural bundles, J. Eur. Math. Soc. (JEMS) 12 (2010), 703–722; MR 2639316(2011e:14036).Google Scholar
Erman, D., The semigroup of Betti diagrams, Algebra Number Theory 3 (2009), 341–365; MR 2525554(2010k:13022).Google Scholar
Erman, D., A special case of the Buchsbaum–Eisenbud–Horrocks rank conjecture, Math. Res. Lett. 17 (2010), 1079–1089; MR 2729632(2012a:13023).Google Scholar
Fløystad, G., Boij–Söderberg theory: introduction and survey, in Progress in commutative algebra 1: combinatorics and homology (De Gruyter, 2012), 1–54.Google Scholar
Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.Google Scholar
McCullough, J., A polynomial bound on the regularity of an ideal in terms of half of the syzygies, Math. Res. Lett. 19 (2012), 555–565.Google Scholar
Sam, S. V. and Weyman, J., Pieri resolutions for classical groups, J. Algebra 329 (2011), 222–259; MR 2769324(2012e:20102).Google Scholar
Schreyer, F.-O. and Eisenbud, D., Betti numbers of syzygies and cohomology of coherent sheaves, in Proceedings of the international congress of mathematicians, Volume II, New Delhi, 2010 (Hindustan Book Agency, 2010), 586–602; MR 2827810.Google Scholar