## 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

Zbl 1095.13500
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