Abstract
We prove regularity bounds for Koszul cycles holding for every ideal of dimension ≤1 in a polynomial ring; see Theorem 3.5. In Theorem 4.7 we generalize the “c+1” lower bound for the Green–Lazarsfeld index of Veronese rings proved in (Bruns et al., arXiv:0902.2431) to the multihomogeneous setting. For the Koszul complex of the c-th power of the maximal ideal in a Koszul ring we prove that the cycles of homological degree t and internal degree ≥t(c+1) belong to the t-th power of the module of 1-cycles; see Theorem 5.2.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L.L. Avramov, A. Conca, S.B. Iyengar, Free resolutions over commutative Koszul algebras. Math. Res. Lett. 17, no. 2 (2010), 197–210
L.L. Avramov and D. Eisenbud, Regularity of modules over a Koszul algebra. J. Algebra 153 (1992), 85–90.
W. Bruns and J. Herzog, Cohen–Macaulay rings. Rev. ed. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press 1998.
W. Bruns, A. Conca and T. Römer, Koszul homology and syzygies of Veronese subalgebras. arXiv:0902.2431, to appear in Math. Ann.
G. Caviglia, Bounds on the Castelnuovo–Mumford regularity of tensor products. Proc. Amer. Math. Soc. 135 (2007), 1949–1957.
A. Conca, J. Herzog, N.V. Trung and G. Valla, Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces. Amer. J. Math. 119 (1997), 859–901.
A. Conca, Regularity jumps for powers of ideals. In: Commutative algebra. Lect. Notes Pure Appl. Math., 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 21–32.
D. Eisenbud, C. Huneke, and B. Ulrich, The regularity of Tor and graded Betti numbers. Amer. J. Math. 128 (2006), 573–605.
S. Eliahou and M. Kervaire, Minimal resolutions of some monomial ideals. J. Algebra 129 (1990), 1–25.
M.L. Green, Koszul cohomology and the geometry of projective varieties. II. J. Differ. Geom. 20 (1984), 279–289.
M. Hering, H. Schenck and G.G. Smith, Syzygies, multigraded regularity and toric varieties. Compos. Math. 142 (2006), 1499–1506.
G. Ottaviani and R. Paoletti, Syzygies of Veronese embeddings. Compos. Math. 125 (2001), 31–37.
B. Sturmfels, Four counterexamples in combinatorial algebraic geometry. J. Algebra 230 (2000), 282–294.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bruns, W., Conca, A., Römer, T. (2011). Koszul Cycles. In: Fløystad, G., Johnsen, T., Knutsen, A. (eds) Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Abel Symposia, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19492-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-19492-4_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19491-7
Online ISBN: 978-3-642-19492-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)