Abstract
We will discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we will see that reg(JM) can be larger than reg(M) + reg(I) even when I is an ideal of linear forms and M is a module with a linear resolution. On the other hand, we have shown that any product of ideals of linear forms has a linear resolution. We will also discuss the case of polymatroidal ideals.
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
D. Bayer, D. Mumford What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991), 1–48, Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993.
A. Capani, G. Niesi, L. Robbiano, CoCoA, a system for doing Computations in Commutative Algebra, Available via ftp from cocoa.dima.unige.it.
K. Chandler, Regularity of the powers of an ideal, Commun.Algebra 25 (1997), 3773–3776.
A. Conca, Straightening law and powers of determinantal ideals of Hankel matrices, Adv. Math. 138 (1998), no. 2, 263–292.
A. Conca, J. Herzog, Castelnuovo-Mumford regularity of products of ideals, preprint 2001.
D. Cutkosky, J. Herzog, N. V. Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity, Compositio Math. 118 (1999), 243–261.
D. Eisenbud, S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133.
A. Geramita, A. Gimigliano, Y. Pitteloud, Graded Betti numbers of some embedded rational n-folds, Math. Ann. 301 (1995), no. 2, 363–380.
J. Herzog, Y. Takayama, Resolutions by mapping cones, to appear in Homology, Homotopy and Applications.
V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), 407–411.
B. Sturmfels, Four counterexamples in combinatorial algebraic geometry, J. Algebra 230 (2000), 282–294.
D. Welsh, Matroid Theory, Academic Press, 1976.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2003 Hindustan Book Agency
About this chapter
Cite this chapter
Conca, A. (2003). Products of Ideals with Linear Resolution. In: Musili, C. (eds) Advances in Algebra and Geometry. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-12-5_1
Download citation
DOI: https://doi.org/10.1007/978-93-86279-12-5_1
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-36-4
Online ISBN: 978-93-86279-12-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)