Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T09:17:44.511Z Has data issue: false hasContentIssue false
  • English
  • Français

Degree bounds on homology and a conjecture of Derksen

Part of: Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  21 September 2016

Marc Chardin  and
Peter Symonds
Marc Chardin
Affiliation:
Institut de Mathématiques de Jussieu, 4, place Jussieu, F-75005 Paris, France email marc.chardin@imj-prg.fr
Peter Symonds
Affiliation:
School of Mathematics, University of Manchester, Manchester M13 9PL, UK email Peter.Symonds@manchester.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Harm Derksen made a conjecture concerning degree bounds for the syzygies of rings of polynomial invariants in the non-modular case [Degree bounds for syzygies of invariants, Adv. Math. 185 (2004), 207–214]. We provide counterexamples to this conjecture, but also prove a slightly weakened version. We also prove some general results that give degree bounds on the homology of complexes and of $\text{Tor}\,$ groups.

Keywords

invariant theorydegree boundslocal cohomology

MSC classification

Primary: 13D02: Syzygies, resolutions, complexes 13A50: Actions of groups on commutative rings; invariant theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 10 , October 2016 , pp. 2041 - 2049
Copyright
© The Authors 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avramov, L. L., Conca, A. and Iyengar, S. B., Free resolutions over commutative Koszul algebras , Math. Res. Lett. 17 (2010), 197–210.Google Scholar
Avramov, L. L., Conca, A. and Iyengar, S. B., Subadditivity of syzygies of Koszul algebras , Math. Ann. 361 (2015), 511–534.CrossRefGoogle Scholar
Broer, A., The direct summand property in modular invariant theory , Transform. Groups 10 (2005), 5–27.Google Scholar
Bruns, W., Conca, A. and Römer, T., Koszul homology and syzygies of Veronese subalgebras , Math. Ann. 351 (2011), 761–779.CrossRefGoogle Scholar
Bruns, W., Conca, A. and Römer, T., Koszul cycles , in Combinatorial aspects of commutative algebra and algebraic geometry, Abel Symposia, vol. 6, eds Fløystad, G., Johnson, T. and Knutsen, A. L. (Springer, Heidelberg, 2011), 17–33.CrossRefGoogle Scholar
Conca, A. and Murai, S., Regularity bounds for Koszul cycles , Proc. Amer. Math. Soc. 143 (2014), 493–503.CrossRefGoogle Scholar
Derksen, H., Degree bounds for syzygies of invariants , Adv. Math. 185 (2004), 207–214.CrossRefGoogle Scholar
Derksen, H. and Sidman, J., A sharp bound for the Castelnuovo–Mumford regularity of subspace arrangements , Adv. Math. 172 (2002), 151–157.CrossRefGoogle Scholar
Fleischmann, P., The Noether bound in invariant theory of finite groups , Adv. Math. 156 (2000), 23–32.CrossRefGoogle Scholar
Fogarty, J., On Noether’s bound for polynomial invariants of a finite group , Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 5–7.CrossRefGoogle Scholar
Snowden, A., A remark on a conjecture of Derksen , J. Commut. Algebra 6 (2014), 109–112.Google Scholar