## Journal of Commutative Algebra

### Cohen-Macaulayness of Rees algebras of diagonal ideals

Kuei-Nuan Lin

#### Abstract

Given two determinantal rings over a field $k$, we consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the secant variety. When the Rees algebra and the symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.

#### Article information

Source
J. Commut. Algebra, Volume 6, Number 4 (2014), 561-586.

Dates
First available in Project Euclid: 5 January 2015

https://projecteuclid.org/euclid.jca/1420466344

Digital Object Identifier
doi:10.1216/JCA-2014-6-4-561

Mathematical Reviews number (MathSciNet)
MR3294862

Zentralblatt MATH identifier
1360.13013

#### Citation

Lin, Kuei-Nuan. Cohen-Macaulayness of Rees algebras of diagonal ideals. J. Commut. Algebra 6 (2014), no. 4, 561--586. doi:10.1216/JCA-2014-6-4-561. https://projecteuclid.org/euclid.jca/1420466344

#### References

• W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambr. Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993.
• S.D. Cutkosky and T. Hà, Arithmetic Macaulayfication of projective schemes, J. Pure Appl. Alg. 201 (2005), 49\textendash61.
• J. Eagon and V. Reiner, Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Alg. 130 (1998), 165–175.
• D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Grad. Texts Math. 150, Springer-Verlag, New York, 1995.
• S. Goto, Y. Nakamura and K. Nishida, Cohen-Macaulay graded rings associated to ideals, Amer. J. Math. 118 (1996), 1197–1213.
• D. Grayson and M. Stillman, Macaulay $2$, A computer algebra system for computing in algebraic geometry and commutative algebra, available through anonymous ftp from http://www.math.uiuc.edu/Macaulay2.
• J. Herzog, A. Simis and W. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), 661\textendash683.
• J. Herzog, A. Simis and W. Vasconcelos, Ideals with sliding depth, Nagoya Math. J. 99 (1985), 159–172.
• T. Hà and N. Trung, Asymptotic behavior of arithmetically Cohen-Macaulay blow-ups, Trans. Amer. Math. Soc. 357 (2005), 3655\textendash3672.
• C. Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), 739\textendash763.
• M. Johnson and B. Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compos. Math. 103 (1996), 7–29.
• K-N Lin, Rees algebras of diagonal ideals, J. Comm. Alg. 5 (2013), 359–398.
• A. Simis and B. Ulrich, On the ideal of an embedded join, J. Algebra 226 (2000), 1–14.
• B. Sturmfels and S. Sullivant, Combinatorial secant varieties, Pure Appl. Math. 2 (2006), 867\textendash891.