zbMATH — the first resource for mathematics

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

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13H05 Regular local rings
13D99 Homological methods in commutative ring theory
Full Text: DOI
[1] Achilles, R; Avramov, L, Relations between properties of a ring and of its associated graded ring, (), 5-29, Band 48
[2] Berman, D, The number of generators of a colength N ideal in a power series ring, J. algebra, 73, 156-166, (1981) · Zbl 0491.13013
[3] Buchsbaum, D; Eisenbud, D, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension three, Amer. J. math., 99, 447-485, (1977) · Zbl 0373.13006
[4] Boratynski, M; Eisenbud, D; Rees, D, On the number of generators of ideals in local Cohen-Macaulay rings, J. algebra, 57, 77-81, (1979) · Zbl 0408.13007
[5] Eisenbud, D; Goto, S, Linear free resolutions and minimal multiplicity, J. algebra, 88, 89-133, (1984) · Zbl 0531.13015
[6] Emsalem, J; Iarrobino, A, Some zero-dimensional generic singularities, Compositio math., 36, 145-188, (1978) · Zbl 0393.14002
[7] Emsalem, J, Géométrie des points épais, Bull. soc. math. France, 106, 399-416, (1978) · Zbl 0396.13017
[8] Fröberg, R; Laksov, D, Compressed algebras, (), 121-151 · Zbl 0558.13007
[9] \scR. Fröberg, Some implications of the spectral sequence Tor^grR(grM, grN)⇒ TorR(M, N), preprint, Stockholm.
[10] Iarrobino, A, Compressed algebras, Trans. amer. math. soc., 285, 337-378, (1984) · Zbl 0548.13009
[11] Iarrobino, A, Compressed algebras and components of the punctual hubert scheme, (), 146-165
[12] Iarrobino, A, Deformations of complete intersections, appendix: Hilbert functions of \(C\)(x, y)I, (), 593-608
[13] Iarrobino, A, Tangent cone of a Gorenstein singularity, (), in press · Zbl 0624.13020
[14] Macaulay, F.S, On a method for dealing with the intersections of two plane curves, Trans. amer. math. soc., 5, 385-410, (1904) · JFM 35.0587.01
[15] Macaulay, F.S, Some properties of enumeration in the theory of modular systems, (), 531-555 · JFM 53.0104.01
[16] Macaulay, F.S, The algebraic theory of modular systems, (1916), Cambridge Univ. Press London/New York · Zbl 0802.13001
[17] Miri, A, Artin modules having extremal Hilbert series: compressed modules, () · Zbl 0784.13006
[18] Nagata, M, Local rings, (1962), Interscience New York · Zbl 0123.03402
[19] Northcott, D.G; Rees, D, Reductions of ideals in local rings, (), 145-158 · Zbl 0057.02601
[20] Robbiano, L; Valla, G, Free resolutions for special tangent cones, () · Zbl 0558.14008
[21] Sally, J, Numbers of generators of ideals in local rings, () · Zbl 0395.13010
[22] Sally, J, Bounds for numbers of generators of Cohen-Macaulay ideals, Pacific J. math., 63, 517-520, (1976) · Zbl 0336.13015
[23] Schenzel, P, Über die freien auflösungen extremaler Cohen-Macaulay ringe, J. algebra, 64, 93-101, (1980) · Zbl 0449.13008
[24] Shalev, A, On the number of generators of ideals in local rings, Adv. in math., 59, 82-94, (1986) · Zbl 0586.13014
[25] Singh, B, Effect of a permissible blowing up on the local Hilbert function, Invent. math., 26, 201-212, (1974) · Zbl 0266.14005
[26] Stanley, R, Hilbert functions of graded algebras, Adv. in math., 28, 57-83, (1978) · Zbl 0384.13012
[27] Watanabe, J, A note on Gorenstein rings of embedding codimension three, Nagoya math. J., 50, 227-232, (1973) · Zbl 0242.13019
[28] Whipple, F, On a theorem due to F. S. Macaulay, J. London math. soc., 8, 431-437, (1928) · JFM 54.0106.18
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.