Hilbert functions and Sally modules. (English) Zbl 0878.13008
Let \((R,m)\) be a local ring of positive Krull dimension, \(I\) an \(m\)-primary ideal, and \(J\) a reduction of \(I\). The Sally module of \(I\) with respect to \(J\) is \(S_JI= \bigoplus_{n\geq 1}I^{n+1}/JI^n\). In this paper the Sally module is studied with the aim of getting information on the Hilbert function \(H_I(n)\) and the associated graded ring \(\text{gr}_I(R)\) of \(R\). In case \(S_J(I)\) can be well understood, this information can be transferred to \(\text{gr}_I(R)\). Here is an example of a result from the paper:
If \(R\) is CM and \(J\) a minimal reduction of \(I\) with reduction number \(r\), then the first Hilbert coefficient of \(S_J(I)\) is \(s_0\leq \sum_{n=1}^{r-1} \lambda(I^{n+1}/JI^n)\). If there is equality, then \(S_J(I)\) is CM, all Hilbert coefficients \(s_j\) of \(S_j(I)\) can be determined, \(\text{depth(gr}_I (R))\geq \dim(R)- 1\), and all Hilbert coefficients of \(\text{gr}_I(R)\) are determined.
If \(R\) is CM and \(J\) a minimal reduction of \(I\) with reduction number \(r\), then the first Hilbert coefficient of \(S_J(I)\) is \(s_0\leq \sum_{n=1}^{r-1} \lambda(I^{n+1}/JI^n)\). If there is equality, then \(S_J(I)\) is CM, all Hilbert coefficients \(s_j\) of \(S_j(I)\) can be determined, \(\text{depth(gr}_I (R))\geq \dim(R)- 1\), and all Hilbert coefficients of \(\text{gr}_I(R)\) are determined.
Reviewer: R.Fröberg (Stockholm)
MSC:
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
| 13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
Keywords:
Cohen-Macaulay ring; Sally module; Hilbert function; associated graded ring; Hilbert coefficientsReferences:
| [1] | Bruns, W.; Herzog, J., Cohen-Macaulay Rings (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0788.13005 |
| [2] | J. Elias, M. E. Rossi, G. Valla, On the coefficients of the Hilbert polynomial; J. Elias, M. E. Rossi, G. Valla, On the coefficients of the Hilbert polynomial · Zbl 0874.13013 |
| [3] | Elias, J.; Valla, G., Rigid Hilbert functions, J. Pure Appl. Algebra, 71, 19-41 (1991) · Zbl 0733.13007 |
| [4] | Guerrieri, A., On the depth of the associated graded ring of an m-primary ideal of a Cohen-Macaulay local ring, J. Algebra, 167, 745-757 (1994) · Zbl 0810.13009 |
| [5] | L. T. Hoa, Two notes on the coefficients of the Hilbert-Samuel polynomial; L. T. Hoa, Two notes on the coefficients of the Hilbert-Samuel polynomial |
| [6] | S. Huckaba, A d-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients, Proc. Amer. Math. Soc.; S. Huckaba, A d-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients, Proc. Amer. Math. Soc. · Zbl 0836.13012 |
| [7] | S. Huckaba, T. Marley, Hilbert coefficients and the depths of associated graded rings; S. Huckaba, T. Marley, Hilbert coefficients and the depths of associated graded rings · Zbl 0910.13008 |
| [8] | Marley, T., The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. (2), 40, 1-8 (1989) · Zbl 0688.13009 |
| [9] | Matsumura, H., Commutative Ring Theory (1986), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0603.13001 |
| [10] | Sally, J. D., On the associated graded ring of a local Cohen-Macaulay ring, J. Math. Kyoto Univ., 17, 19-21 (1977) · Zbl 0353.13017 |
| [11] | Sally, J. D., Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra, 56, 168-183 (1979) · Zbl 0401.13016 |
| [12] | Sally, J. D., Tangent cones at Gorenstein singularities, Compositio Math., 40, 167-175 (1980) · Zbl 0389.13009 |
| [13] | Sally, J. D., Hilbert coefficients and reduction number 2, J. Algebraic Geom., 1, 325-333 (1992) · Zbl 0779.13005 |
| [14] | Sally, J. D., Ideals whose Hilbert function and Hilbert polynomial agree at \(n\), J. Algebra, 157, 534-547 (1993) · Zbl 0780.13008 |
| [15] | Vasconcelos, W. V., Hilbert functions, analytic spread, and Koszul homology, Contemp. Math., 159, 401-422 (1994) · Zbl 0803.13012 |
| [16] | M. Vaz Pinto, 1995, Structure of Sally Modules and Hilbert Functions, Rutgers University; M. Vaz Pinto, 1995, Structure of Sally Modules and Hilbert Functions, Rutgers University |
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