Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals


Author: Lê Tuân Hoa
Journal: Trans. Amer. Math. Soc. 360 (2008), 4519-4540
MSC (2000): Primary 13D45, 13D40, 13P10
DOI: https://doi.org/10.1090/S0002-9947-08-04424-3
Published electronically: April 4, 2008
MathSciNet review: 2403695
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated with reduced algebras over an algebraically closed field with a given arithmetic degree and dimension. A good bound is also given for the Castelnuovo-Mumford regularity of initial ideals which depends neither on term orders nor on the coordinates and holds for any field.


References [Enhancements On Off] (What's this?)

  • [BM] D. Bayer and 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. MR 95d:13032
  • [BS] D. Bayer and M. Stillman, A criterion for detecting $ m$-regularity, Invent. Math. 87(1987), no. 1, 1-11. MR 87k:13019
  • [BEL] A. Bertram, L. Ein and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4(1991), 587-602. MR 92g:14014
  • [Bl] C. Blancafort, Hilbert functions of graded algebras over Artinian rings, J. Pure Appl. Algebra 125(1998), no. 1-3, 55-78. MR 98m:13023
  • [BrL] M. P. Brodmann and A. F. Lashgari, A diagonal bound for cohomological postulation numbers of projective schemes, J. Algebra 265(2003), 631-650. MR 2004f:14030
  • [BrS] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics, 60. Cambridge University Press, Cambridge, 1998. MR 99h:13020
  • [CS] G. Caviglia and E. Sbarra, Characteristic-free bounds for Castelnuovo-Mumford regularity, Compos. Math. 141 (2005), no. 6, 1365-1373. MR 2006i:13032
  • [CF] Marc Chardin and Amadou Lamine Fall, Sur la régularité de Castelnuovo-Mumford des idéaux, en dimension 2, C. R. Math. Acad. Sci. Paris 341 (2005), no. 4, 233–238 (French, with English and French summaries). MR 2164678, https://doi.org/10.1016/j.crma.2005.06.020
  • [CM] M. Chardin and G. Moreno-Socias, Regularity of lex-segment ideals: Some closed formulas and applications, Proc. Amer. Math. Soc. 131(2003), no. 4, 1093-1102. MR 2003m:13014
  • [E] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. MR 97a:13001
  • [EG] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88(1984), no. 1, 89-133. MR 85f:13023
  • [FOV] H. Flenner, L. O'Carroll and W. Vogel, Joins and intersections. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999. MR 2001b:14010
  • [Gi] M. Giusti, Some effectivity problems in polynomial ideal theory, EUROSAM 84 (Cambridge, 1984) Lecture Notes in Comput. Sci., vol. 174, Springer, Berlin, 1984, pp. 159–171. MR 779123, https://doi.org/10.1007/BFb0032839
  • [GLP] L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72(1983), no. 3, 491-506. MR 85g:14033
  • [HPV] 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, https://doi.org/10.1090/conm/331/05909
  • [HH] Lê Tuân Hoa and Eero Hyry, Castelnuovo-Mumford regularity of initial ideals, J. Symbolic Comput. 38 (2004), no. 5, 1327–1341. MR 2168718, https://doi.org/10.1016/j.jsc.2004.04.001
  • [HSV] Le Tuan Hoa, Jürgen Stückrad, and Wolfgang Vogel, Towards a structure theory for projective varieties of 𝑑𝑒𝑔𝑟𝑒𝑒=𝑐𝑜𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛+2, J. Pure Appl. Algebra 71 (1991), no. 2-3, 203–231. MR 1117635, https://doi.org/10.1016/0022-4049(91)90148-U
  • [HT] L. T. Hoa; N. V. Trung, On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals, Math. Z. 229(1998), no. 3, 519-537. MR 99k:13034
  • [K] S. L. Kleiman, Les théorèmes de finitude pour le foncteur de Picard, in: ``Théorie des intersections et théorème de Riemann-Roch'' (French) Séminaire de Géométrie Algébrique du Bois-Marie 1966-1967 (SGA 6), pp. 616-666. Lecture Notes in Mathematics, Vol. 225. Springer-Verlag, Berlin-New York, 1971. MR 50:7133
  • [MVY] C. Miyazaki, W. Vogel and K. Yanagawa, Associated primes and arithmetic degrees, J. Algebra 192(1997), no. 1, 166-182. MR 98i:13036
  • [MM] H. M. Möller and F. Mora, Upper and lower bounds for the degree of Gröbner bases, EUROSAM 84 (Cambridge, 1984), 172-183, Lecture Notes in Comput. Sci., 174, Springer, Berlin, 1984. MR 86k:13008
  • [M] D. Mumford, Lectures on curves on an algebraic surface, Princeton Univ. Press, Princeton, 1966. MR 35:187
  • [RTV1] M. E. Rossi, N. V. Trung and G. Valla, Castelnuovo-Mumford regularity and extended degree, Trans. Amer. Math. Soc. 355(2003), no. 5, 1773-1786. MR 2004b:13020
  • [RTV2] Maria Evelina Rossi, Ngô Viêt Trung, and Giuseppe Valla, Castelnuovo-Mumford regularity and finiteness of Hilbert functions, Commutative algebra, Lect. Notes Pure Appl. Math., vol. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 193–209. MR 2184798, https://doi.org/10.1201/9781420028324.ch14
  • [RVV] M. E. Rossi, G. Valla, and W. V. Vasconcelos, Maximal Hilbert functions, Results Math. 39 (2001), no. 1-2, 99–114. MR 1817403, https://doi.org/10.1007/BF03322678
  • [S] E. Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29(2001), no. 12, 5383-5409. MR 2002j:13024
  • [Sj] R. Sjögren, On the regularity of graded $ k$-algebras of Krull dimension $ \le 1$, Math. Scand. 71(1992), 167-172. MR 94b:13010
  • [V] W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry. With chapters by D. Eisenbud, D. R. Grayson, J. Herzog and M. Stillman. Algorithms and Computation in Mathematics, 2. Springer-Verlag, Berlin, 1998. MR 99c:13048

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D45, 13D40, 13P10

Retrieve articles in all journals with MSC (2000): 13D45, 13D40, 13P10


Additional Information

Lê Tuân Hoa
Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
Email: lthoa@math.ac.vn

DOI: https://doi.org/10.1090/S0002-9947-08-04424-3
Keywords: Castelnuovo-Mumford regularity, local cohomology, Hilbert function, Hilbert polynomial, initial ideal.
Received by editor(s): July 13, 2005
Published electronically: April 4, 2008
Additional Notes: The author was supported in part by the National Basic Research Program (Vietnam). The final preparation of the article was done during his stay at the Centre de Recerca Matematica (Spain).
Dedicated: Dedicated to Professor J. Herzog on the occasion of his 65th birthday
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society