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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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