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The Hilbert function of a Cohen-Macaulay local algebra: Extremal Gorenstein algebras. (English) Zbl 0628.13016
Let R,M,k be a regular local ring with $$R\geq k$$ and assume k is infinite. Let $$A=R/I,m$$ be a Cohen-Macaulay quotient of R. For any R-module B, H(B) denotes the associated Hilbert series $$\sum \ell (M^ iB/M^{i+1}B)z^ i.$$ The socle type $$E(A)$$ is defined as the series $$H(0:_{\bar A}\bar m)$$, where $$\bar A,\bar m$$ is the quotient of $$A,m$$ by an ideal generated by a generic maximal linear A-sequence. A is said to be compressed when E(A) has a certain minimal property.
The first of the two main results states that $$H(A)$$ is majorized by the Hilbert series of a compressed algebra of the same dimension and codimension and socle type determined by E(A). This extends from the graded to the ungraded case a result of R. Fröberg and D. Laksov [in Complete intersections, Lect. 1st Sess. C.I.M.E., Acireale/Italy 1983, Lect. Notes Math. 1092, 121-151 (1984; Zbl 0558.13007)]. - The second theorem requires A to be Gorenstein and connects by sharp inequalities the order d(I) (the largest d such that $$M^ d\supseteq I)$$, the multiplicity m(A) and the socle degree j(A) (the largest j such that $$\bar m^ j\neq 0)$$. In particular Schenzel’s inequality $$j\geq 2d-2$$ [P. Schenzel, J. Algebra 64, 93-101 (1980; Zbl 0449.13008)] is extended from the graded to the ungraded case. When $$j=2d-2$$, A is said to be extremal and, in this case, A has an Artinian quotient $$\bar A$$ which is compressed with even socle degree. Finally by using the structure theorem of D. A. Buchsbaum and D. Eisenbud [Am. J. Math. 99, 447-485 (1977; Zbl 0373.13006)] for Gorenstein ideals of height three, the authors establish inequalities for m(A) in terms of the minimal number of generators of I when A has codimension three.
Reviewer: D.Kirby

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13H05 Regular local rings 13D99 Homological methods in commutative ring theory
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