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Transactions of the American Mathematical Society

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The homological degree of a module

Author: Wolmer V. Vasconcelos
Journal: Trans. Amer. Math. Soc. 350 (1998), 1167-1179
MSC (1991): Primary 13D40; Secondary 13D45, 13P10
MathSciNet review: 1458335
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Abstract: A homological degree of a graded module $M$ is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when $M$ is not Cohen-Macaulay. We construct a degree, $\operatorname{hdeg}(M)$, that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo-Mumford's regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.

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  • 1. Bayer, D., Mumford, D., What can be computed in Algebraic Geometry? D. Eisenbud et al. (eds.), Computational Algebraic Geometry and Commutative Algebra, pp. 1-48, Cortona 1991, Cambridge: Cambridge University Press, 1993.MR 95d:13032
  • 2. Bayer, D., Stillman, M., Macaulay, a system for computation in algebraic geometry and commutative algebra, 1992. Available via anonymous ftp from
  • 3. Bruns, W., Herzog, J., Cohen-Macaulay Rings, Cambridge University Press, Cambridge, 1993MR 95h:13020
  • 4. Capani, A., Niesi, G., Robbiano, L. CoCoA, a system for doing computations in commutative algebra, 1995. Available via anonymous ftp from
  • 5. Eakin, P., Sathaye, A., Prestable ideals, J. Algebra 41 (1976), 439-454 MR 54:7449
  • 6. Hartshorne, R., Connectedness of the Hilbert scheme, Publications Math. I.H.E.S. 29 (1966), 261-304. MR 35:4232
  • 7. Miyazaki, C., Vogel, W., Towards a theory of arithmetic degree, Manuscripta Math. 89 (1996), 427-438. MR 97f:14003
  • 8. Sally, J.D., Bounds for numbers of generators for Cohen-Macaulay ideals, Pacific J. Math. 63 (1976), 517-520. MR 53:13208
  • 9. Sally. J.D., Numbers of Generators of Ideals in Local Rings, Lecture Notes in Pure & Applied Math. 35. Marcel Dekker, New York, 1978. MR 58:5654
  • 10. Stückrad, J., Vogel, W., Buchsbaum Rings and Applications, Springer, Vienna-New York, 1986. MR 88h:13011
  • 11. Sturmfels, B., Trung, N.V., Vogel, W., Bounds on degrees of projective schemes, Math. Annalen 302 (1995), 417-432. MR 96i:13029
  • 12. Ngo Viet Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229-236. MR 89i:13031
  • 13. Ngo Viet Trung, Bounds for the minimum number of generators of generalized Cohen-Macaulay ideals, J. Algebra 90 (1984), 1-9. MR 85h:13012
  • 14. Valla, G., Generators of ideals and multiplicities, Comm. in Algebra 9 (1981), 1541-1549. MR 83d:13030
  • 15. Vasconcelos, W.V., Arithmetic of Blowup Algebras, London Math. Soc., Lecture Note Series 195, Cambridge University Press, Cambridge, 1994. MR 95g:13005
  • 16. Vasconcelos, W.V., The reduction number of an algebra, Compositio Math. 104 (1996), 189-197. CMP 97:04

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Additional Information

Wolmer V. Vasconcelos
Affiliation: Department of Mathematics - Hill Center, Rutgers University, 110 Frelinghuysen RD, Piscataway, New Jersey 08854-8019

Keywords: Arithmetic degree, Castelnuovo-Mumford regularity, geometric degree, Gorenstein ring, homological degree, reduction number
Received by editor(s): June 3, 1996
Additional Notes: The author was partially supported by the NSF
Article copyright: © Copyright 1998 American Mathematical Society

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