## 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

### Citations:

Zbl 0628.13012; Zbl 0849.13009; Zbl 0688.13009
Full Text: