A 𝑑-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients

Huckaba, Sam

doi:10.1090/S0002-9939-96-03182-6

A $d$-dimensional extension of a lemma of Huneke’s and formulas for the Hilbert coefficients
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Proc. Amer. Math. Soc. 124 (1996), 1393-1401 Request permission

Abstract:

A $d$-dimensional version is given of a $2$-dimensional result due to C. Huneke. His result produced a formula relating the length $\lambda (I^{n+1}/JI^{n})$ to the difference $P_{I}(n+1)-H_{I}(n+1)$, where $I$ is primary for the maximal ideal of a $2$-dimensional Cohen-Macaulay local ring $R$, $J$ is a minimal reduction of $I$, $H_{I}(n)=\lambda (R/I^{n})$, and $P_{I}(n)$ is the Hilbert-Samuel polynomial of $I$. We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of $I$. We also include a characterization, in terms of the Hilbert coefficients of $I$, of the condition $depth(G(I))\geq d-1$.

References

W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993.
A. Guerrieri, On the depth of certain graded rings associated to an ideal, Ph.D. Dissertation, Purdue University (1993).
Craig Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318. MR 894879, DOI 10.1307/mmj/1029003560
David Kirby and Hefzi A. Mehran, A note on the coefficients of the Hilbert-Samuel polynomial for a Cohen-Macaulay module, J. London Math. Soc. (2) 25 (1982), no. 3, 449–457. MR 657501, DOI 10.1112/jlms/s2-25.3.449
Kazuji Kubota, On the Hilbert-Samuel function, Tokyo J. Math. 8 (1985), no. 2, 439–448. MR 826998, DOI 10.3836/tjm/1270151224
T. Marley, Hilbert functions of ideals in Cohen-Macaulay local rings, Ph.D. Dissertation, Purdue University (1989).
Thomas Marley, The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. (2) 40 (1989), no. 1, 1–8. MR 1028910, DOI 10.1112/jlms/s2-40.1.1
Masayoshi Nagata, Local rings, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Corrected reprint. MR 0460307
Masao Narita, A note on the coefficients of Hilbert characteristic functions in semi-regular local rings, Proc. Cambridge Philos. Soc. 59 (1963), 269–275. MR 146212, DOI 10.1017/S0305004100036884
D. G. Northcott, A note on the coefficients of the abstract Hilbert function, J. London Math. Soc. 35 (1960), 209–214. MR 110731, DOI 10.1112/jlms/s1-35.2.209
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Akira Ooishi, $\Delta$-genera and sectional genera of commutative rings, Hiroshima Math. J. 17 (1987), no. 2, 361–372. MR 909621
Judith D. Sally, Hilbert coefficients and reduction number $2$, J. Algebraic Geom. 1 (1992), no. 2, 325–333. MR 1144442
Judith D. Sally, Ideals whose Hilbert function and Hilbert polynomial agree at $n=1$, J. Algebra 157 (1993), no. 2, 534–547. MR 1220782, DOI 10.1006/jabr.1993.1114
Irena Swanson, A note on analytic spread, Comm. Algebra 22 (1994), no. 2, 407–411. MR 1255875, DOI 10.1080/00927879408824857
Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101. MR 514892, DOI 10.1017/S0027763000018225
Wolmer V. Vasconcelos, Hilbert functions, analytic spread, and Koszul homology, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 401–422. MR 1266195, DOI 10.1090/conm/159/01520
Yinghwa Wu, Reduction numbers and Hilbert polynomials of ideals in higher-dimensional Cohen-Macaulay local rings, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 1, 47–56. MR 1131477, DOI 10.1017/S0305004100075149
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876