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The Hilbert Coefficients of the Fiber Cone and the a-Invariant of the Associated Graded Ring

Published online by Cambridge University Press:  20 November 2018

Clare D'Cruz  and
Tony J. Puthenpurakal
Clare D'Cruz
Affiliation:
Chennai Mathematical Institute, Plot H1, SIPCOT IT Park Padur PO, Siruseri 603103, India, e-mail: clare@cmi.ac.in
Tony J. Puthenpurakal
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India, e-mail: tputhen@math.iitb.ac.in
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Abstract

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Let $(A,\mathfrak{m})$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I)={{\oplus }_{n\ge 0}}{{I}^{n}}/\mathfrak{m}{{I}^{n}}$ be the fiber cone of $I$. We prove certain relations among the Hilbert coefficients ${{f}_{0\,}}(I),\,{{f}_{1}}(I),\,{{f}_{2}}(I)$ of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

Keywords

13A3013D40fiber conea-invariantHilbert coefficients of fiber cone
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 4 , 01 August 2009 , pp. 762 - 778
Copyright
Copyright © Canadian Mathematical Society 2012

References

[1] Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications. Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998.Google Scholar
[2] Bruns, W. and Herzog, J., Cohen- Macaulay rings. Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.Google Scholar
[3] Capani, A., Niesi, G., and Robbiano, L., CoCoA, a System for Doing Computations in Commutative Algebra, 1995, available via anonymous ftp from cocoa.dima.unige.it.Google Scholar
[4] Corso, A., Sally modules of m-primary ideals in local rings. arXiv:math.AC/0309027.Google Scholar
[5] Corso, A., Polini, C., and Rossi, M. E., Depth of associated graded rings via Hilbert coefficients of ideals. J. Pure Appl. Algebra 201(2005), no. 1-3, 126–141.Google Scholar
[6] Corso, A., Polini, C., and Vasconcelos, W. V., Multiplicity of the special fiber of blowups. Math. Proc. Cambridge Philos. Soc. 140(2006), no. 2, 207–219.Google Scholar
[7] Cortadellas, T. and Zarzuela, S., On the depth of the fiber cone of filtrations. J. Algebra 198(1997), no. 2, 428–445.Google Scholar
[8] D’Cruz, C. and Verma, J. K., Hilbert series of fiber cones of ideals with almost minimal mixed multiplicity. J. Algebra 251(2002), no. 1, 98–109.Google Scholar
[9] Elias, J., Depth of higher associated graded rings. J. London Math. Soc. 70(2004), no. 1, 41–58.Google Scholar
[10] Hoa, L. T., Reduction numbers and Rees algebras of powers of an ideal. Proc. Amer. Math. Soc. 119(1993), no. 2, 415–422.Google Scholar
[11] Huckaba, S. and Marley, T., On associated graded rings of normal ideals. J. Algebra 222(1999), 146–163.Google Scholar
[12] Huneke, C., Hilbert functions and symbolic powers. Michigan Math. J. 34(1987), no. 2, 293–318.Google Scholar
[13] Itoh, S., Integral closures of ideals generated by regular sequences. J. Algebra 117(1988), no. 2, 390–401.Google Scholar
[14] Jayanthan, A. V., Puthenpurakal, T. J., and Verma, J. K., On fiber cones of m-primary ideals. Canad. J. Math 59(2007), no. 1, 109–126.Google Scholar
[15] Jayanthan, A. V. and Verma, J. K., Hilbert coefficients and depth of fiber cones. J. Pure Appl. Algebra 201(2005), no. 1-3, 97–115.Google Scholar
[16] Marley, T., The coefficients of the Hilbert polynomial and the reduction number of an ideal. J. London Math. Soc. 40(1989), no. 1, 1–8.Google Scholar
[17] Narita, M., A note on the coefficients of Hilbert characteristic functions in semi-regular local rings. Proc. Cambridge Philos. Soc. 59(1963), 269–275.Google Scholar
[18] Northcott, D. G., A note on the coefficients of the abstract Hilbert function. J. London Math. Soc. 35(1960), 209–214.Google Scholar
[19] Northcott, D. G. and Rees, D., Reductions of ideals in local rings. Proc. Cambridge Philos. Soc. 50 (1954), 145–158.Google Scholar
[20] Puthenpurakal, T. J., Hilbert-coefficients of a Cohen- Macaulay module, J. Algebra 264(2003), no. 1, 82–97.Google Scholar
[21] Puthenpurakal, T. J., Invariance of a length associated to a reduction. Comm. Algebra 33(2005), no. 6, 2039–2042.Google Scholar
[22] Puthenpurakal, T. J., Ratliff-Rush filtration, regularity and depth of higher associated graded modules. I. J. Pure Appl. Algebra 208(2007), no. 1, 159–176.Google Scholar
[23] Rees, D., Generalizations of reductions and mixed multiplicities. J. London Math. Soc. 29(1984), no. 3, 397–414.Google Scholar
[24] Sally, J. D., Tangent cones at Gorenstein singularities. Compositio Math. 40(1980), no. 2, 167–175.Google Scholar
[25] Shah, K., On the Cohen- Macaulayness of the fiber cone of an ideal. J. Algebra 143(1991), no. 1, 156–172.Google Scholar
[26] Trung, N. V., Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101(1987), no. 2, 229–236.Google Scholar
[27] Valabrega, P. and Valla, G., Form rings and regular sequences. Nagoya Math. J. 72(1978), 93–101.Google Scholar
[28] Valla, G., Problems and results on Hilbert functions of graded algebras. In: Six Lectures on Commutative Algebra, Progr. Math. 166. Birkhäuser, Basel, 1998, pp. 293–344.Google Scholar