## Kyoto Journal of Mathematics

### On the almost Gorenstein property in the Rees algebras of contracted ideals

#### Abstract

We explore the question of when the Rees algebra $\mathcal{R}(I)=\bigoplus_{n\ge0}I^{n}$ of $I$ is an almost Gorenstein graded ring, where $R$ is a 2-dimensional regular local ring and $I$ is a contracted ideal of $R$. Recently, we showed that $\mathcal{R}(I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present article show that if $I$ is a contracted ideal with $\mathrm{o}(I)\le2$, then $\mathcal{R}(I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I)\ge3$, then $\mathcal{R}(I)$ is not necessarily an almost Gorenstein graded ring, even though $I$ is a contracted stable ideal. Thus both affirmative and negative answers are given.

#### Article information

Source
Kyoto J. Math., Volume 59, Number 4 (2019), 769-785.

Dates
Revised: 1 June 2017
Accepted: 8 June 2017
First available in Project Euclid: 24 October 2019

https://projecteuclid.org/euclid.kjm/1571904147

Digital Object Identifier
doi:10.1215/21562261-2018-0001

Mathematical Reviews number (MathSciNet)
MR4032199

Zentralblatt MATH identifier
07193997

#### Citation

Goto, Shiro; Matsuoka, Naoyuki; Taniguchi, Naoki; Yoshida, Ken-ichi. On the almost Gorenstein property in the Rees algebras of contracted ideals. Kyoto J. Math. 59 (2019), no. 4, 769--785. doi:10.1215/21562261-2018-0001. https://projecteuclid.org/euclid.kjm/1571904147

#### References

• [1] V. Barucci and R. Fröberg, One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), 418–442.
• [2] J. P. Brennan, J. Herzog, and B. Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), 181–203.
• [3] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge Univ. Press, Cambridge, 1993.
• [4] S. Goto, Integral closedness of complete-intersection ideals, J. Algebra 108 (1987), 151–160.
• [5] S. Goto, N. Matsuoka, and T. T. Phuong, Almost Gorenstein rings, J. Algebra 379 (2013), 355–381.
• [6] S. Goto, N. Matsuoka, N. Taniguchi, and K.-I. Yoshida, The almost Gorenstein Rees algebras of parameters, J. Algebra 452 (2016), 263–278.
• [7] S. Goto, N. Matsuoka, N. Taniguchi, and K.-I. Yoshida, The almost Gorenstein Rees algebras over 2-dimensional regular local rings, J. Pure Appl. Algebra 220 (2016), 3425–3436.
• [8] S. Goto and Y. Shimoda, “On the Rees algebras of Cohen-Macaulay local rings” in Commutative Algebra (Fairfax, Va., 1979), Lect. Notes Pure Appl. Math. 68, Dekker, New York, 1982, 201–231.
• [9] S. Goto, R. Takahashi, and N. Taniguchi, Almost Gorenstein rings—towards a theory of higher dimension, J. Pure Appl. Algebra 219 (2015), 2666–2712.
• [10] J. Herzog, “Die Struktur des kanonischen Moduls: Anwendungen” in Der kanonische Modul eines Cohen-Macaulay-Rings (Regensburg, 1970/1971), Lecture Notes in Math. 238, Springer, Berlin, 1971, 59–84.
• [11] J. Herzog, Certain complexes associated to a sequence and a matrix, Manuscripta Math. 12 (1974), 217–248.
• [12] C. Huneke, “Complete ideals in 2-dimensional regular local rings” in Commutative Algebra (Berkeley, Calif., 1987), Math. Sci. Res. Inst. Publ. 15, Springer, New York, 1989, 325–338.
• [13] C. Huneke and J. D. Sally, Birational extensions in dimension 2 and integrally closed ideals, J. Algebra 115 (1988), 481–500.
• [14] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge Univ. Press, Cambridge, 2006.
• [15] J. Lipman, “On complete ideals in regular local rings” in Algebraic Geometry and Commutative Algebra, Vol. I, Kinokuniya, Tokyo, 1987, 203–231.
• [16] F. Muiños and F. Planas-Vilanova, The equations of Rees algebras of equimultiple ideals of deviation one, Proc. Amer. Math. Soc. 141 (2013), 1241–1254.
• [17] D. Rees, Hilbert functions and pseudorational local rings of dimension 2, J. Lond. Math. Soc. (2) 24 (1981), 467–479.
• [18] J. D. Sally, Cohen-Macaulay local rings of maximal embedding dimension, J. Algebra 56 (1979), 168–183.
• [19] B. Ulrich, Ideals having the expected reduction number, Amer. J. Math. 118 (1996), 17–38.
• [20] J. K. Verma, Joint reductions and Rees algebras, Math. Proc. Cambridge Philos. Soc. 109 (1991), 335–342.
• [21] J. K. Verma, Rees algebras of contracted ideals in 2-dimensional regular local rings, J. Algebra 141 (1991), 1–10.
• [22] O. Zariski, Polynomial ideals defined by infinitely near base points, Amer. J. Math. 60 (1938), 151–204.
• [23] O. Zariski and P. Samuel, Commutative Algebra, II, Van Nostrand, Princeton, 1960.