The Hilbert function of a Cohen-Macaulay local algebra: Extremal Gorenstein algebras.

*(English)*Zbl 0628.13016Let 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.

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 |

##### Keywords:

regular local ring; Cohen-Macaulay quotient; Hilbert series of a compressed algebra; Gorenstein; order; multiplicity; socle degree
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\textit{J. Elias} and \textit{A. Iarrobino}, J. Algebra 110, 344--356 (1987; Zbl 0628.13016)

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