|
Extensions of the Multiplicity conjecture
Authors:
Juan Migliore, Uwe Nagel and Tim Römer
Journal:
Trans. Amer. Math. Soc. 360 (2008), 2965-2985
MSC (2000):
Primary 13H15, 13D02; Secondary 13C40, 14M12, 13C14, 14H50
Posted:
November 28, 2007
MathSciNet review:
2379783
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The Multiplicity conjecture of Herzog, Huneke, and Srinivasan states an upper bound for the multiplicity of any graded  -algebra as well as a lower bound for Cohen-Macaulay algebras. In this note we extend this conjecture in several directions. We discuss when these bounds are sharp, find a sharp lower bound in the case of not necessarily arithmetically Cohen-Macaulay one-dimensional schemes of 3-space, and propose an upper bound for finitely generated graded torsion modules. We establish this bound for torsion modules whose codimension is at most two.
- 1.
Annetta
Aramova and Jürgen
Herzog, Almost regular sequences and Betti numbers, Amer. J.
Math. 122 (2000), no. 4, 689–719. MR 1771569
(2001i:13029)
- 2.
D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp.
- 3.
Christopher
A. Francisco, New approaches to bounding the multiplicity of an
ideal, J. Algebra 299 (2006), no. 1,
309–328. MR 2225778
(2007b:13043), http://dx.doi.org/10.1016/j.jalgebra.2005.06.004
- 4.
Christopher
A. Francisco and Hema
Srinivasan, Multiplicity conjectures, Syzygies and Hilbert
functions, Lect. Notes Pure Appl. Math., vol. 254, Chapman &
Hall/CRC, Boca Raton, FL, 2007, pp. 145–178. MR 2309929
(2008f:13044), http://dx.doi.org/10.1201/9781420050912.ch5
- 5.
Leah
H. Gold, A degree bound for codimension two lattice ideals, J.
Pure Appl. Algebra 182 (2003), no. 2-3,
201–207. MR 1903053
(2004i:13023), http://dx.doi.org/10.1016/S0022-4049(02)00324-9
- 6.
L.H. Gold, H. Schenck and H. Srinivasan, Betti numbers and degree bounds for some linked zero-schemes. To appear in Canad. J. Math.
- 7.
Elena
Guardo and Adam
Van Tuyl, Powers of complete intersections: graded Betti numbers
and applications, Illinois J. Math. 49 (2005),
no. 1, 265–279 (electronic). MR 2157379
(2006k:13035)
- 8.
Jürgen
Herzog and Hema
Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2879–2902. MR 1458304
(99g:13033), http://dx.doi.org/10.1090/S0002-9947-98-02096-0
- 9.
Jürgen
Herzog and Xinxian
Zheng, Notes on the multiplicity conjecture, Collect. Math.
57 (2006), no. 2, 211–226. MR 2223853
(2007a:13029)
- 10.
Craig
Huneke and Matthew
Miller, A note on the multiplicity of Cohen-Macaulay algebras with
pure resolutions, Canad. J. Math. 37 (1985),
no. 6, 1149–1162. MR 828839
(87d:13024), http://dx.doi.org/10.4153/CJM-1985-062-4
- 11.
Craig
Huneke and Bernd
Ulrich, General hyperplane sections of algebraic varieties, J.
Algebraic Geom. 2 (1993), no. 3, 487–505. MR 1211996
(94b:14046)
- 12.
Jan
O. Kleppe, Juan
C. Migliore, Rosa
Miró-Roig, Uwe
Nagel, and Chris
Peterson, Gorenstein liaison, complete intersection liaison
invariants and unobstructedness, Mem. Amer. Math. Soc.
154 (2001), no. 732, viii+116. MR 1848976
(2002e:14083)
- 13.
Juan
C. Migliore, Introduction to liaison theory and deficiency
modules, Progress in Mathematics, vol. 165, Birkhäuser
Boston Inc., Boston, MA, 1998. MR 1712469
(2000g:14058)
- 14.
Juan
C. Migliore, Submodules of the deficiency module, J. London
Math. Soc. (2) 48 (1993), no. 3, 396–414. MR 1241777
(94i:14051), http://dx.doi.org/10.1112/jlms/s2-48.3.396
- 15.
J.
Migliore and U.
Nagel, Monomial ideals and the Gorenstein liaison class of a
complete intersection, Compositio Math. 133 (2002),
no. 1, 25–36. MR 1918287
(2003g:13010), http://dx.doi.org/10.1023/A:1016358116986
- 16.
Juan
Migliore, Uwe
Nagel, and Tim
Römer, The multiplicity conjecture in low codimensions,
Math. Res. Lett. 12 (2005), no. 5-6, 731–747.
MR
2189234 (2006i:13042)
- 17.
Rosa
M. Miró-Roig, A note on the multiplicity of determinantal
ideals, J. Algebra 299 (2006), no. 2,
714–724. MR 2228336
(2007a:13030), http://dx.doi.org/10.1016/j.jalgebra.2005.05.017
- 18.
P. Rao, Liaison among Curves in
 , Invent. Math. 50 (1979), 205-217.
- 19.
Tim
Römer, Note on bounds for multiplicities, J. Pure Appl.
Algebra 195 (2005), no. 1, 113–123. MR 2100313
(2005g:13041), http://dx.doi.org/10.1016/j.jpaa.2004.05.008
- 20.
Hema
Srinivasan, A note on the multiplicities of Gorenstein
algebras, J. Algebra 208 (1998), no. 2,
425–443. MR 1655460
(99m:13046), http://dx.doi.org/10.1006/jabr.1998.7413
- 1.
- A. Aramova and J. Herzog, Almost regular sequences and Betti numbers. Amer. J. Math. 122 (2000), 689-719. MR 1771569 (2001i:13029)
- 2.
- D. Bayer and M. Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra. Source and object code available for Unix and Macintosh computers. Contact the authors, or download from ftp://math.harvard.edu via anonymous ftp.
- 3.
- C. Francisco, New approaches to bounding the multiplicity of an ideal, J. Algebra 299 (2006), no. 1, 309-328. MR 2225778 (2007b:13043)
- 4.
- C. Francisco and H. Srinivasan, Multiplicity conjectures, Syzygies and Hilbert Functions 145-178, Lecture Notes Pure Appl. Math., 254, Chapman & Hall/CRC Boca Raton, FL, 2007. MR 2309929
- 5.
- L.H. Gold, A degree bound for codimension two lattice ideals, J. Pure Appl. Algebra 183 (2003), 201-207. MR 1903053 (2004i:13023)
- 6.
- L.H. Gold, H. Schenck and H. Srinivasan, Betti numbers and degree bounds for some linked zero-schemes. To appear in Canad. J. Math.
- 7.
- E. Guardo and A. Van Tuyl, Powers of complete intersections: Graded Betti numbers and applications. Ill. J. Math. 49 (2005), no.1, 265-279. MR 2157379 (2006k:13035)
- 8.
- J. Herzog and H. Srinivasan, Bounds for multiplicities. Trans. Amer. Math. Soc. 350 (1998), no. 7, 2879-2902. MR 1458304 (99g:13033)
- 9.
- J. Herzog and Zheng, Notes on the multiplicity conjecture, Collect. Math 57 (2006), no. 2, 211-226. MR 2223853 (2007a:13029)
- 10.
- C. Huneke and M. Miller, A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions, Canad. J. Math. 37 (1985), 1149-1162. MR 828839 (87d:13024)
- 11.
- C. Huneke and B. Ulrich, General Hyperplane Sections of Algebraic Varieties, J. Alg. Geom. 2 (1993), 487-505. MR 1211996 (94b:14046)
- 12.
- J. Kleppe, J. Migliore, R.M. Miró-Roig, U. Nagel, and C. Peterson, Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness, Memoirs of the Amer. Math. Soc. 154, 2001; 116 pp. MR 1848976 (2002e:14083)
- 13.
- J. Migliore, ``Introduction to Liaison Theory and Deficiency Modules,'' Progress in Mathematics 165, Birkhäuser, 1998. MR 1712469 (2000g:14058)
- 14.
- J. Migliore, Submodules of the deficiency module, J. London Math. Soc. 48(3) (1993), 396-414. MR 1241777 (94i:14051)
- 15.
- J. Migliore and U. Nagel, Monomial Ideals and the Gorenstein Liaison Class of a Complete Intersection, Compositio Math. 133 (2002), 25-36. MR 1918287 (2003g:13010)
- 16.
- J. Migliore, U. Nagel and T. Römer, The multiplicity conjecture in low codimensions. Math. Res. Lett. 12 (2005), No. 5-6, 731-747. MR 2189234 (2006i:13042)
- 17.
- R. Miró-Roig, A note on the multiplicity of determinantal ideals, J. Algebra 299 (2006), 714-724. MR 2228336 (2007a:13030)
- 18.
- P. Rao, Liaison among Curves in , Invent. Math. 50 (1979), 205-217.
- 19.
- T. Römer, Note on bounds for multiplicities, J. Pure Appl. Algebra 195 (2005), 113-123. MR 2100313 (2005g:13041)
- 20.
- H. Srinivasan, A note on the multiplicities of Gorenstein algebras, J. Algebra 208 (1998), no. 2, 425-443. MR 1655460 (99m:13046)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
13H15,
13D02,
13C40,
14M12,
13C14,
14H50
Retrieve articles in all journals
with MSC (2000):
13H15,
13D02,
13C40,
14M12,
13C14,
14H50
Additional Information
Juan Migliore
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email:
Juan.C.Migliore.1@nd.edu
Uwe Nagel
Affiliation:
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
Email:
uwenagel@ms.uky.edu
Tim Römer
Affiliation:
FB Mathematik/Informatik, Universität Osnabrück, 49069 Osnabrück, Germany
Email:
troemer@uos.de
DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04360-7
PII:
S 0002-9947(07)04360-7
Received by editor(s):
March 2, 2006
Posted:
November 28, 2007
Additional Notes:
Part of the work for this paper was done while the first author was sponsored by the National Security Agency under Grant Number MDA904-03-1-0071
Article copyright:
© Copyright 2007 American Mathematical Society
|
