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A sharp bound for the minimal number of generators of perfect height two ideals. (English) Zbl 0588.13018
Let (R,M) be a regular local ring, and let I be a perfect ideal of R - i.e., the factor ring $$A=R/I$$ is Cohen-Macaulay. The author observes that if I has height h and $$I\subset M^ t$$, then $$e(A)=(h^{t-1+h})+\ell (M^ t/(I+(X.)M^{t-1}))$$ where e(A) is the multiplicity of A and (X.) is the preimage in M of a minimal reduction of M/I in A. Therefore, if $$h=2$$ and v(I) denotes the minimal number of generators of I, then $$v(I)(v(I)-1)\leq 2e(A).$$
Examples are given of height two perfect ideals I for which this is an equality and several characterizations are given of such ideals I involving the existence of a standard basis for I of elements of order v(I)-1, and properties of the Hilbert function of R/I.
Reviewer: W.Heinzer

##### MSC:
 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13H05 Regular local rings 13H15 Multiplicity theory and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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##### References:
  BURCH, L.: On ideals of finite homological dimension in local rings. Math.Proc.Camb.Phil.Soc. 64,941-946 (1968) · Zbl 0172.32302 · doi:10.1017/S0305004100043620  EAGON, J.A. and NORTHCOTT, D.G.: Ideals defined by matrices and a certain complex associated with them. Proc.Royal Soc.Set.A 269,188-204(1962) · Zbl 0106.25603 · doi:10.1098/rspa.1962.0170  ELIAS, J. and IARROBINO, A.: Extremal Gorenstein algebras of codimension three; the Hilbert function of a Cohen-Macaulay local algebra. Preprint 1984 · Zbl 0628.13016  MACAULAY; F. S. The algebraic theory of modular systems. Cambridge Univ.Press, 1916 · JFM 46.0167.01  MOH, T. T.: On generators of ideals. Proc.Amer.Math. Soc. 77,309-312(1979) · Zbl 0449.13001 · doi:10.1090/S0002-9939-1979-0545586-0  NORTHCOTT, D.G. and REES, D.: Reductions of ideals in local rings. Math.Proc.Camb.Phil.Soc. 50,145-158 (1954) · Zbl 0057.02601 · doi:10.1017/S0305004100029194  ORECCHIA, F.: Gli esempi di Macaulay. Quaderni C.N.R. (1981)  ROBBIANO, L. and VALLA, G.: Free resolutions of special tangent cones.Commutative Algebra.Proceedings of the Trento Conference.Lect.Notes in Pure and Appl. Math.Series,84.Marcel Dekker(l983) · Zbl 0558.14008  SALLY, J.D.: Number of generators of ideals in local rings.Lect.Notes in Pure and Appl.Math.Series, 35.Marcel Dekker(1978) · Zbl 0395.13010  VALLA, G.: Generators of ideals and multiplicities. Communications in Alg.l5(1981) · Zbl 0525.13015  VALLABREGA, P. and VALLA, G.: Form rings and regular sequences, Nagoya Math.J. 72,93-101(1978) · Zbl 0362.13007  VUT TRUNG, N.: Bounds for the minimum number of generators of generalised Cohen-Macaulay rings. Journal of Alg.90(1984)
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