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Postulation and reduction vectors of multigraded filtrations of ideals. (English) Zbl 1390.13003

To any graded ring one can assign a Hilbert function \(H\) and a Hilbert polynomial \(P\). For example, let \(I\) be a primary ideal of \(2\)-dimensional Cohen-Macaulay local ring and \(J\) be a minimal reduction of \(I\). In [Mich. Math. J. 34, 293–318 (1987; Zbl 0628.13012)] C. Huneke proved a lemma relating \(l(I^{n+1}/JI^n)\) to the difference \(P_I(n+1)-H_I(n+1)\). S. Huckaba [Proc. Am. Math. Soc. 124, No. 5, 1393–1401 (1996; Zbl 0849.13009)] extended this result to the higher-dimensional case. Under some conditions involving the depth of an associated graded ring of \(I\), T. Marley [J. Lond. Math. Soc., II. Ser. 40, No. 1, 1–8 (1989; Zbl 0688.13009)] proved:
a) if \(e_i=0\) then \(e_j=0\) for all \(j>i\) (here, \(e_i\) is the \(i\)-th Hilbert coefficient of \(I\)).
b) \(r(I) = n(I) + d\) (here, \(r(I)\) is the reduction number of \(I\), \(n(I)\) is the postulation number of \(I\) and \(d\) is the dimension of \(R\)).
The paper under review extends the mentioned results to the corresponding properties of Hilbert functions and Hilbert polynomials of multigraded filtrations of ideals. Also, a relationship between the Cohen-Macaulay property of the multigraded Rees algebra with respect to multigraded filtration and the reduction vectors are given.

MSC:

13A02 Graded rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13D45 Local cohomology and commutative rings
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