Skip to main content Accessibility help

On Special Fiber Rings of Modules

  • Cleto B. Miranda-Neto (a1)

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$ -module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$ , our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$ , which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

Hide All

The author was partially supported by CAPES-Brazil (grant 88881.121012/2016-01), and by CNPq-Brazil (grant 421440/2016-3).

Hide All
[1] Aberbach, I. M. and Huneke, C., An improved Briançon-Skoda theorem with applications to the Cohen–Macaulayness of Rees algebras . Math. Ann. 297(1993), 343369.
[2] Brennan, J., Ulrich, B., and Vasconcelos, W. V., The Buchsbaum-Rim polynomial of a module . J. Algebra 241(2001), 379392.
[3] Bruns, W. and Herzog, J., Cohen–Macaulay rings. Revised Edition. Cambridge University Press, Cambridge, 1998.
[4] Buchsbaum, D. and Rim, D. S., A generalized Koszul complex. II. Depth and multiplicity . Trans. Amer. Math. Soc. 111(1964), 197224.
[5] Chan, C.-Y. J., Liu, J.-C., and Ulrich, B., Buchsbaum-Rim multiplicities as Hilbert-Samuel multiplicities . J. Algebra 319(2008), 44134425.
[6] Corso, A., Ghezzi, L., Polini, C., and Ulrich, B., Cohen–Macaulayness of special fiber rings . Comm. Algebra 31(2003), 37133734.
[7] Corso, A., Polini, C., and Vasconcelos, W., Multiplicity of the special fiber of blowups . Math. Proc. Cambridge Philos. Soc. 140(2006), 207219.
[8] Cortadellas, T. and Zarzuela, S., On the structure of the fiber cone of ideals with analytic spread one . J. Algebra 317(2007), 759785.
[9] D’ Cruz, C. and Verma, J. K., Hilbert series of fiber cones of ideals with almost minimal mixed multiplicity . J. Algebra 251(2002), 98109.
[10] Eisenbud, D. and Huneke, C., Cohen–Macaulay Rees algebras and their specialization . J. Algebra 81(1983), 202224.
[11] Eisenbud, D., Huneke, C., and Ulrich, B., What is the Rees algebra of a module? Proc. Amer. Math. Soc. 131(2002), 701708.
[12] Goto, S., Hayasaka, F., Kurano, K., and Nakamura, Y., Rees algebras of the second syzygy module of the residue field of a regular local ring . Contemp. Math. 390(2005), 97108.
[13] Heinzer, W. and Kim, M.-K., Properties of the fiber cone of ideals in local rings . Comm. Algebra 31(2003), 35293546.
[14] Huckaba, S. and Huneke, C., Rees algebras of ideals having small analytic deviation . Trans. Amer. Math. Soc. 339(1993), 373402.
[15] Huckaba, S. and Marley, T., Depth properties of Rees algebras and associated graded rings . J. Algebra 156(1993), 259271.
[16] Huckaba, S. and Marley, T., On associated graded rings of normal ideals . J. Algebra 222(1999), 146163.
[17] Huneke, C., On the associated graded ring of an ideal . Illinois J. Math. 26(1982), 121137.
[18] Huneke, C. and Sally, J., Birational extensions in dimension two and integrally closed ideals . J. Algebra 115(1988), 481500.
[19] Huneke, C. and Swanson, I., Integral closure of ideals, rings and modules. London Math. Soc. Lecture Note Ser., 336. Cambridge University Press, Cambridge, 2006.
[20] Jayanthan, A. V., Puthenpurakal, T. J., and Verma, J. K., On fiber cones of  $\mathfrak{m}$ -primary ideals. Canad. J. Math. 59(2007), 109–126.
[21] Korb, T. and Nakamura, Y., On the Cohen–Macaulayness of multi-Rees algebras and Rees algebras of powers of ideals . J. Math. Soc. Japan 50(1998), 451467.
[22] Kurano, K., On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal . J. Algebra 190(1997), 405434.
[23] Lima, P. H. and Jorge Pérez, V. H., On the Gorenstein property of the fiber cone to filtration . Int. J. Algebra 8(2014), 159174.
[24] Lin, K.-N. and Polini, C., Rees algebras of truncations of complete intersections . J. Algebra 410(2014), 3652.
[25] Lipman, J., Cohen–Macaulayness in graded algebras . Math. Res. Lett. 1(1994), 149157.
[26] Miranda-Neto, C. B., Graded derivation modules and algebraic free divisors . J. Pure Appl. Algebra 219(2015), 54425466.
[27] Miranda-Neto, C. B., On Aluffi’s problem and blowup algebras of certain modules . J. Pure Appl. Algebra 221(2017), 799820.
[28] Ooishi, A., On the Gorenstein property of the associated graded ring and the Rees algebra of an ideal . J. Algebra 155(1993), 397414.
[29] Polini, C. and Ulrich, B., Necessary and sufficient conditions for the Cohen–Macaulayness of blowup algebras . Compos. Math. 119(1999), 185207.
[30] Polini, C. and Xie, Y., j-multiplicity and depth of associated graded modules . J. Algebra 379(2013), 3149.
[31] Sancho de Salas, J. B., Blowing-up morphisms with Cohen–Macaulay associated graded rings . In: Géométrie algébrique et applications, I. Travaux en Cours, 22. Hermann, Paris, 1987, pp. 201209.
[32] Shah, K., On the Cohen–Macaulayness of the fiber cone of an ideal . J. Algebra 143(1991), 156172.
[33] Simis, A., Ulrich, B., and Vasconcelos, W., Rees algebras of modules . Proc. London Math. Soc. 87(2003), 610646.
[34] Trung, N. V. and Ikeda, S., When is the Rees algebra Cohen–Macaulay? Comm. Algebra 17(1989), 28932922.
[35] Trung, N. V., Viet, D. Q., and Zarzuela, S., When is the Rees algebra Gorenstein? J. Algebra 175(1995), 137156.
[36] Vasconcelos, W. V., Arithmetic of blowup algebras. London Math. Soc. Lecture Note Ser., 195, Cambridge University Press, Cambridge, 1994.
[37] Vasconcelos, W. V., Integral closure. Rees algebras, multiplicities, algorithms. Springer Monographs on Mathematics. Springer-Verlag, Berlin, 2005.
[38] Viet, D. Q., On the multiplicity and the Cohen–Macaulayness of fiber cones of graded algebras . J. Pure Appl. Algebra 213(2009), 21042116.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed