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The number of generators of some homogeneous ideals. (English) Zbl 0831.13005

Let \(k\) be an infinite field, let \(A = \bigoplus^\infty_{i = 0} A_i\) be a graded Cohen-Macaulay ring with \(A_0 = k\), and assume \(A\) is generated as a \(k\)-algebra by \(A_1 \). Let \(M = \bigoplus^\infty_{i = 1} A_i\) and let \(e\) be the multiplicity of \(A\). For a regular homogeneous ideal \(I\) of \(A\) let \(\mu (I)\) denote the minimal number of homogeneous elements needed to generate \(I\). Let \(j(I) = \min \{t \geq 1 \mid I \cap A_t\) contains a regular element}, and \(l(I) = \min \{t \geq 1 \mid M^t \subseteq I\}\).
The main result is that if \(A\) has dimension one then \(\mu (I) \leq e + j(I) - \max \{r, l(I)\}\). This improves the standard result \(\mu (I) \leq e\), and includes a conjecture of A. V. Geramita as a special case. As an application the author gives a new proof of a result of P. Dubreil on the minimum number of generators of a perfect, height two, homogeneous ideal in a polynomial ring \(k[X_0, \ldots, X_n]\).

MSC:

13E15 Commutative rings and modules of finite generation or presentation; number of generators
13A15 Ideals and multiplicative ideal theory in commutative rings
13A02 Graded rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

[1] Ciliberto C., Bollettino U.M.I 1 pp 633– (1987)
[2] Davis E.D., Bull. Sci. Math 108 pp 143– (1984)
[3] Dubreil P., Bull. Soc. Math. t 61 pp 258– (1933)
[4] Geramita A.V., Bull. Sc. Math. 2e series 107 pp 193– (1983)
[5] DOI: 10.1016/0001-8708(78)90045-2 · Zbl 0384.13012 · doi:10.1016/0001-8708(78)90045-2
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