## Journal of Commutative Algebra

### A Northcott type inequality for Buchsbaum-Rim coefficients

#### Abstract

In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that, if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

#### Article information

Source
J. Commut. Algebra, Volume 8, Number 4 (2016), 493-512.

Dates
First available in Project Euclid: 27 October 2016

https://projecteuclid.org/euclid.jca/1477600744

Digital Object Identifier
doi:10.1216/JCA-2016-8-4-493

Mathematical Reviews number (MathSciNet)
MR3566527

Zentralblatt MATH identifier
1358.13007

#### Citation

Balakrishnan, R.; Jayanthan, A.V. A Northcott type inequality for Buchsbaum-Rim coefficients. J. Commut. Algebra 8 (2016), no. 4, 493--512. doi:10.1216/JCA-2016-8-4-493. https://projecteuclid.org/euclid.jca/1477600744

#### References

• P.B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambr. Philos. Soc. 53 (1957), 568–575.
• J. Brennan, B. Ulrich and W.V. Vasconcelos, The Buchsbaum-Rim polynomial of a module, J. Algebra 241 (2001), 379–392.
• D.A. Buchsbaum and D.S. Rim, A generalized Koszul complex II, Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224.
• F. Hayasaka and E. Hyry, A family of graded modules associated to a module, Comm. Algebra 36 2008, 4201–4217.
• ––––, On the Buchsbaum-Rim function of a parameter module, J. Algebra 327 (2011), 307–315.
• C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293–318.
• C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, Lond. Math. Soc. Lect. Notes 336, Cambridge University Press, Cambridge, 2006.
• E. Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), 2213–2232.
• ––––, Cohen-Macaulay multi-Rees algebras, Comp. Math. 130 (2002), 319–343.
• A.V. Jayanthan and J.K. Verma, Grothendieck-Serre formula and bigraded Cohen-Macaulay Rees algebra, J. Algebra 254 (2002), 1–20.
• D. Katz and V. Kodiyalam, Symmetric powers of complete modules over a two dimensional regular local ring, Trans. Amer Math. Soc. 349 (1997), 747–762.
• J.-C. Liu, Rees algebras of finitely generated torsion free modules over a two dimensional regular local ring, Comm. Algebra 26 (1998), 4015–4039.
• D.G. Northcott, A note on the coefficients of the abstract Hilbert function, J. Lond. Math. Soc. 35 (1960), 209–214.
• A. Ooishi, $\Delta$-genera and sectional genera of commutative rings, Hiroshima Math. J. 17 (1987), 361–372.
• P. Roberts, Multiplicities and Chern characters in local algebra, Cambridge University Press, New York, 1998.
• A. Simis, B. Ulrich and W. Vasconcelos, Rees algebras of modules, Proc. Lond. Math. Soc. 87 (2003), 610–646.
• W. Vasconcelos, Integral closure: Rees algebra, multiplicities, algorithms, Springer Monographs in Mathematics, Springer Verlag, New York, 2005.
• J.K. Verma, Joint reductions of complete ideals, Nagoya Math. J. 118 (1990), 155–163.