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On complete d-sequences and the defining ideals of Rees algebras

Published online by Cambridge University Press:  24 October 2008

Sam Huckaba
Sam Huckaba
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306-3027, U.S.A.
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Abstract

If R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 3 , November 1989 , pp. 445 - 458
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

REFERENCES

[1]Barshay, J.. Graded algebras of powers of ideals generated by A-sequences. J. Algebra 25 (1973), 90–99.CrossRefGoogle Scholar
[2]Cowsik, R. C. and Nori, M. V.. On the fibres of blowing up. J. Indian Math. Soc. 40 (1976), 217–222.Google Scholar
[3]Eisenbud, D. and Huneke, C.. Cohen-Macaulay Rees algebras and their specialization. J. Algebra 81 (1983), 202–224.CrossRefGoogle Scholar
[4]Goto, S. and Shimoda, Y.. On the Rees algebras of Cohen-Macaulay local rings. In Commutative Algebra (Analytic Methods), Lecture Notes in Pure and Appl. Math. no. 68 (Marcel Dekker Inc., 1982), pp. 202–231.Google Scholar
[5]Goto, S. and Shimoda, Y.. On the Gorensteinness of Rees and form rings of almost complete intersections. Nagoya Math. J. 92 (1983), 69–88.CrossRefGoogle Scholar
[6]Herzog, J., Simis, A. and Vasconcelos, W. V.. Koszul homology and blowing up rings. In Proceedings Trento Commutative Algebra Conference, Lecture Notes in Pure and Appl. Math. no. 84 (Marcel Dekker Inc., 1983), pp. 79–169.Google Scholar
[7]Herzog, J., Simis, A. and Vasconcelos, W. V.. On the arithmetic and homology of algebras of linear type. Trans. Amer. Math. Soc. 277 (1983), 739–763.Google Scholar
[8]Hochster, M.. Criteria for equality of ordinary and symbolic powers of primes. Math. Z. 133 (1973), 53–65.CrossRefGoogle Scholar
[9]Huckaba, S.. Reduction numbers for ideals of higher analytic spread. Math. Proc. Cambridge Philos. Soc. 102 (1987), 49–57.CrossRefGoogle Scholar
[10]Huckaba, S.. Symbolic powers of prime ideals with applications to hypersurface rings. Nagoya Math. J. 113 (1989), 161–172.CrossRefGoogle Scholar
[11]Huckaba, S.. Analytic spread modulo an element and symbolic Rees algebras. J. Algebra (to appear).Google Scholar
[12]Huneke, C.. On the symmetric and Rees algebra of an ideal generated by a d-sequence. J. Algebra 62 (1980), 268–275.CrossRefGoogle Scholar
[13]Huneke, C.. On the associated graded ring of an ideal. Illinois J. Math. 26 (1982), 121–137.CrossRefGoogle Scholar
[14]Huneke, C.. The theory of d-sequences and powers of ideals. Adv. in Math. 46 (1982), 249–279.CrossRefGoogle Scholar
[15]Huneke, C.. Determinatal ideals of linear type. Arch. Math. (Basel) 47 (1986), 324–329.CrossRefGoogle Scholar
[16]Matsumura, H.. Commutative Algebra. 2nd ed. (Benjamin/Cummings, 1980).Google Scholar
[17]Micali, A.. Sur les algébres universelles. Ann. Inst. Fourier (Grenoble) 14 (1964), 33–88.CrossRefGoogle Scholar
[18]Nagata, M., Local Rings (Krieger, 1975).Google Scholar
[19]Northcott, D. G. and Rees, D.. Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50(1954), 145–158.CrossRefGoogle Scholar
[20]Trung, N. V.. Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (1987), 229–236.CrossRefGoogle Scholar
[21]Valla, G.. On the symmetric and Rees algebras of an ideal. Manuscripta Math. 30 (1980), 239–255.CrossRefGoogle Scholar
[22]Vasconcelos, W. V.. Koszul homology and the structure of low codimension Cohen–Macaulay ideals. Trans. Amer. Math. Soc. 301 (1987), 591–613.CrossRefGoogle Scholar