The Dedekind-Mertens formula

and determinantal rings

Authors:
Winfried Bruns and Anna Guerrieri

Journal:
Proc. Amer. Math. Soc. **127** (1999), 657-663

MSC (1991):
Primary 13C40, 13C14, 13D40, 13P10

MathSciNet review:
1468185

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank Cohen-Macaulay modules over the determinantal rings .

**1.**D. Bayer and M. Stillman.*Macaulay: a system for computation in algebraic geometry and commutative algebra*. Available by anonymous`ftp`from`zariski.harvard.edu`.**2.**G. Boffi, W. Bruns, and A. Guerrieri.*On the jacobian ideal of a trilinear form.*Preprint.**3.**Winfried Bruns and Jürgen Herzog,*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956****4.**Winfried Bruns and Udo Vetter,*Determinantal rings*, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR**953963****5.**Aldo Conca and Jürgen Herzog,*On the Hilbert function of determinantal rings and their canonical module*, Proc. Amer. Math. Soc.**122**(1994), no. 3, 677–681. MR**1213858**, 10.1090/S0002-9939-1994-1213858-0**6.**A. Corso, W. V. Vasconcelos, and R. Villareal.*Generic Gaussian Ideals*. J. Pure Appl. Algebra, to appear.**7.**David Eisenbud,*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960****8.**S. Glaz and W. V. Vasconcelos.*The content of Gaussian polynomials*. J. Algebra, to appear**9.**W. Heinzer and C. Huneke.*The Dedekind-Mertens Lemma and the contents of polynomials*. Proc. Amer. Math. Soc., to appear.**10.**W. Heinzer and C. Huneke.*Gaussian polynomials and content ideals*. Proc. Amer. Math. Soc., to appear.**11.**Jürgen Herzog and Ngô Viêt Trung,*Gröbner bases and multiplicity of determinantal and Pfaffian ideals*, Adv. Math.**96**(1992), no. 1, 1–37. MR**1185786**, 10.1016/0001-8708(92)90050-U

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Additional Information

**Winfried Bruns**

Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

Email:
Winfried.Bruns@mathematik.uni-osnabrueck.de

**Anna Guerrieri**

Affiliation:
Universität Osnabrück, FB Mathematik/Informatik, 49069 Osnabrück, Germany

Email:
guerran@univaq.it

DOI:
https://doi.org/10.1090/S0002-9939-99-04535-9

Keywords:
Dedekind--Mertens formula,
initial ideal,
determinantal ring,
Cohen--Macaulay module

Received by editor(s):
January 22, 1997

Received by editor(s) in revised form:
June 16, 1997

Additional Notes:
The visit of the first author to the University of L’Aquila that made this paper possible was supported by the Vigoni program of the DAAD and the CRUI

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1999
American Mathematical Society