Extended degree functions and monomial modules

Nagel, Uwe; Römer, Tim

doi:10.1090/S0002-9947-05-03848-1

Extended degree functions and monomial modules
HTML articles powered by AMS MathViewer

by Uwe Nagel and Tim Römer PDF

Trans. Amer. Math. Soc. 358 (2006), 3571-3589 Request permission

Abstract:

The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.

References

Dave Bayer and David Mumford, What can be computed in algebraic geometry?, Computational algebraic geometry and commutative algebra (Cortona, 1991) Sympos. Math., XXXIV, Cambridge Univ. Press, Cambridge, 1993, pp. 1–48. MR 1253986
D. Bayer, The division algorithm and the Hilbert scheme, Ph.D. thesis, Harvard University, 1982.
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it.
Nguyen Tu Cuong and Le Thanh Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra 267 (2003), no. 1, 156–177. MR 1993472, DOI 10.1016/S0021-8693(03)00225-4
Luisa Rodrigues Doering, Tor Gunston, and Wolmer V. Vasconcelos, Cohomological degrees and Hilbert functions of graded modules, Amer. J. Math. 120 (1998), no. 3, 493–504. MR 1623400, DOI 10.1353/ajm.1998.0019
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
D. R. Grayson, M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
G. M. Greuel, G. Pfister, H. Schönemann, Singular 2.0 A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). Available at http://www.singular.uni-kl.de.
T. Gunston, Cohomological degrees, Dilworth numbers and linear resolutions. Ph.D. Thesis, Rutgers University, 1998.
Jürgen Herzog and Enrico Sbarra, Sequentially Cohen-Macaulay modules and local cohomology, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 327–340. MR 1940671
Jürgen Herzog, Dorin Popescu, and Marius Vladoiu, On the Ext-modules of ideals of Borel type, Commutative algebra (Grenoble/Lyon, 2001) Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 171–186. MR 2013165, DOI 10.1090/conm/331/05909
Chikashi Miyazaki and Wolfgang Vogel, Towards a theory of arithmetic degrees, Manuscripta Math. 89 (1996), no. 4, 427–438. MR 1383523, DOI 10.1007/BF02567527
Uwe Nagel, Characterization of some projective subschemes by locally free resolutions, Commutative algebra (Grenoble/Lyon, 2001) Contemp. Math., vol. 331, Amer. Math. Soc., Providence, RI, 2003, pp. 235–266. MR 2013169, DOI 10.1090/conm/331/05913
Uwe Nagel, Non-degenerate curves with maximal Hartshorne-Rao module, Math. Z. 244 (2003), no. 4, 753–773. MR 2000458, DOI 10.1007/s00209-003-0520-4
Uwe Nagel, Comparing Castelnuovo-Mumford regularity and extended degree: the borderline cases, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3585–3603. MR 2146640, DOI 10.1090/S0002-9947-04-03595-0
Uwe Nagel, Roberto Notari, and Maria Luisa Spreafico, Curves of degree two and ropes on a line: their ideals and even liaison classes, J. Algebra 265 (2003), no. 2, 772–793. MR 1987029, DOI 10.1016/S0021-8693(03)00237-0
Maria Evelina Rossi, Ngô Viêt Trung, and Giuseppe Valla, Castelnuovo-Mumford regularity and extended degree, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1773–1786. MR 1953524, DOI 10.1090/S0002-9947-03-03185-4
Enrico Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29 (2001), no. 12, 5383–5409. MR 1872238, DOI 10.1081/AGB-100107934
Enrico Sbarra, Ideals with maximal local cohomology modules, Rend. Sem. Mat. Univ. Padova 111 (2004), 265–275. MR 2076743
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Jürgen Stückrad and Wolfgang Vogel, Buchsbaum rings and applications, Springer-Verlag, Berlin, 1986. An interaction between algebra, geometry and topology. MR 881220, DOI 10.1007/978-3-662-02500-0
Bernd Sturmfels, Ngô Viêt Trung, and Wolfgang Vogel, Bounds on degrees of projective schemes, Math. Ann. 302 (1995), no. 3, 417–432. MR 1339920, DOI 10.1007/BF01444501
Wolmer V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR 1484973, DOI 10.1007/978-3-642-58951-5
Wolmer V. Vasconcelos, The homological degree of a module, Trans. Amer. Math. Soc. 350 (1998), no. 3, 1167–1179. MR 1458335, DOI 10.1090/S0002-9947-98-02127-8
Ken-ichi Yoshida, A generalization of linear Buchsbaum modules in terms of homological degree, Comm. Algebra 26 (1998), no. 3, 931–945. MR 1606198, DOI 10.1080/00927879808826174