Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Betti numbers of modules of essentially monomial type

Author: Shou-Te Chang
Journal: Proc. Amer. Math. Soc. 128 (2000), 1917-1926
MSC (1991): Primary 13D25, 18G10; Secondary 13H05
Published electronically: February 25, 2000
MathSciNet review: 1653433
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $R$ be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of $R$-modules of essentially monomial type. These modules are defined with respect to various $R$-regular sequences. For example, finite length modules of monomial type over regular local rings of dimension $n$ are modules of essentially monomial type with respect to $R$-regular sequences of length $n$. If a module is of essentially monomial type with respect to an $R$-regular sequence of length $n$, then the rank of its $i$-th syzygy is at least $\binom {n-1}{i-1}$ and its $i$-th Betti number is at least $\binom ni$.

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

  • 1. D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (3) (1977), 447-485. MR 56:11983
  • 2. -, Generic free resolutions and a family of generically perfect fields, Adv. in Math. 18 (1975), 245-301. MR 53:391
  • 3. -, What makes a complex exact, J. of Algebra (1973), 259-268. MR 47:3369
  • 4. S.-T. Chang, Betti numbers of modules of exponent two over regular local rings, J. of Algebra 193 (1997), 640-659. CMP 97:15
  • 5. H. Charalambous, Lower bounds for betti numbers of multigarded modules, J. of Algebra 137 (1991), 491-500. MR 92b:13020
  • 6. H. Charalambous and E. G. Evans, Problems on Betti Numbers of finite length modules, Free resolutions in Commutative Algebra and Algebraic Geometry (D. Eisenbud and C. Huneke, eds.), Research Notes in Mathematics, vol. 2, Jones and Bartlett, Boston, 1992, pp. 25-33. MR 93e:13018
  • 7. H. Charalambous, E. G. Evans and M. Miller, Betti numbers for modules of finite legnth, Proc. of the A. M. S. 109 (1990), 63-70. MR 90j:13021
  • 8. D. Dugger, Betti numbers of almost complete intersections, preprint.
  • 9. E. G. Evans and P. Griffith, Syzygies, London Math. Soc. Lecture Note Series, Cambridge University Press, 1985. MR 87b:13001
  • 10. -, Binomial behavior of Betti numbers for modules of finite length, Pacific J. of Math. 133 (1988), 267-276. MR 89d:13014
  • 11. R. Hartshorne, Algebraic vector bundles on projective spaces: a problem list, Topology 18 (1979), 117-128. MR 81m:14014
  • 12. J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 13-14 (1984), 1627-1646. MR 85e:13021
  • 13. G. Horrocks, Vector bundles on the punctured spectrum of a regular local ring, Proc. London Math. Soc. (3) 14 (1964), 689-713. MR 30:120
  • 14. C. Huneke and B. Ulrich, The structure of linkage, Annals of Math. 126 (1987), 277-334. MR 88k:13020
  • 15. L. Santoni, Horrocks' question for monomially graded modules, Pacific J. of Math. 141, (1) (1990), 105-124. MR 91b:13022
  • 16. J.-P. Serre, Algèbre locale. Multiplicités, Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New York, 1965. MR 34:1352

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13D25, 18G10, 13H05

Retrieve articles in all journals with MSC (1991): 13D25, 18G10, 13H05

Additional Information

Shou-Te Chang
Affiliation: Department of Mathematics, National Chung Cheng University, Minghsiung, Chiayi 621, Taiwan, R.O.C.

Received by editor(s): March 24, 1998
Received by editor(s) in revised form: September 1, 1998
Published electronically: February 25, 2000
Additional Notes: The author is partially supported by an N.S.C. grant of R.O.C
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society