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Monomial ideals and \(n\)-lists. (English) Zbl 1087.13012

Generalizing a construction of A. V. Geramita, T. Harima and Y. S. Shin [Ill. J. Math 45, 1–23 (2001; Zbl 1095.13500)], the author introduces so-called \(n\)-lists: A \(1\)-list is a natural number, and for \(n\geq 1\) an \(n\)-list is a decreasing infinite sequence of \((n- 1)\)-lists, where \(A\geq B\) for two \(n\)-lists \(A= (A_i)\) and \(B= (B_i)\) if \(A_i\geq B_i\) for all \(i\). Next an injective map \(\Phi\) from the set of \(n\)-lists to the set of monomial ideals in the polynomial ring \(k[x_1,\dots, x_n]\) (\(k\) a field) is defined. (\(\Phi\) is not surjective which does’nt affect the applications given by the author. In a final section the author shows how one may generalize the set of \(n\)-lists to make \(\Phi\) surjective.)
The map \(\Phi\) serves to characterize the Artinian monomial ideals in terms of \(n\)-lists. Furthermore, it is used to calculate the multiplicities and Hilbert polynomial degrees for the quotients of Borel fixed ideals, and to give a new proof of the result due to Geramita, Harima, and Shin (loc. cit.).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H15 Multiplicity theory and related topics
13A15 Ideals and multiplicative ideal theory in commutative rings

Citations:

Zbl 1095.13500
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