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An inequality for Hilbert series of graded algebras. (English) Zbl 0582.13007
Let $$R=k[X_ 1,...,X_ n]/I$$, where k is a field and $$I=(f_ 1,...,f_ r)$$ is a homogeneous ideal. Let $$d=\dim R$$, $$g=depth R$$ and $$d_ i=\deg f_ i$$. We define $$Hilb_ R(Z)=\sum (\dim_ kR_ i)z^ i,$$ where $$R=\oplus R_ i$$ is the natural decomposition of R into graded pieces. -
Theorem. Hilb$$_ R(Z)\geq (1/(1-Z^ g))\max ([\prod^{r}_{i=1}(1- Z^{d_ i})/(1-Z)^{n-g}],1/(1-Z)^{d-g}),$$ where $$[\sum a_ iZ^ i]=\sum b_ iZ^ i$$ with $$b_ i=a_ i$$ if $$a_ j>0$$ for $$i\leq j$$ and $$b_ i=0$$ otherwise and $$\max(A(Z),B(Z)) = \sum_{l\geq 0}\max (a_ i,b_ i)Z$$.
Call R extremal if the above inequality is an equality. We give lots of examples of extremal rings and conjecture that if R is Cohen-Macaulay there exists an extremal ring with the same numerical characters as R. Two classes of extremal almost complete intersections are studied more closely with respect to their Koszul homology and Poincaré series.
Note. The conjecture has now been confirmed by D. Anick (Mass. Inst. Technology), for $$n=3$$.

##### MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13H15 Multiplicity theory and related topics 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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