## Illinois Journal of Mathematics

### Coefficient ideals in dimension two

A. Kohlhaas

#### Abstract

We describe coefficient ideals for both $(x,y)$-primary monomial ideals in $k[x,y]$ and $\mathfrak{m}$-primary ideals in two-dimensio-nal regular local rings $(R,\mathfrak{m})$ by linking them to certain ideals of reduction number one. In the monomial case, we then explicitly determine the generators of a coefficient ideal by showing their symmetric relationship to the generators of the associated reduction number one ideal.

#### Article information

Source
Illinois J. Math., Volume 58, Number 4 (2014), 1041-1053.

Dates
Revised: 27 March 2015
First available in Project Euclid: 6 November 2015

https://projecteuclid.org/euclid.ijm/1446819300

Digital Object Identifier
doi:10.1215/ijm/1446819300

Mathematical Reviews number (MathSciNet)
MR3421598

Zentralblatt MATH identifier
1327.13020

#### Citation

Kohlhaas, A. Coefficient ideals in dimension two. Illinois J. Math. 58 (2014), no. 4, 1041--1053. doi:10.1215/ijm/1446819300. https://projecteuclid.org/euclid.ijm/1446819300

#### References

• I. Aberbach and A. Hosry, The Briançon–Skoda theorem and coefficient ideals for non-$\mathfrak{m}$-primary ideals, Proc. Amer. Math. Soc. 139 (2011), no. 11, 3903–3907.
• I. Aberbach and C. Huneke, A theorem of Briançon–Skoda type for regular local rings containing a field, Proc. Amer. Math. Soc. 124 (1996), 707–713.
• W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Univ. Press, Cambridge, 1993.
• A. Corso, C. Polini and B. Ulrich, The structure of the core of ideals, Math. Ann. 321 (2001), 89–105.
• A. Corso, C. Polini and B. Ulrich, Core and residual intersections of ideals, Trans. Amer. Math. Soc. 354 (2002), 2579–2594.
• A. Corso, C. Polini and B. Ulrich, Core of projective dimension one modules, Manuscripta Math. 111 (2003), 427–433.
• S. Goto and Y. Shimoda, On the Rees algebras of Cohen–Macaulay local rings, Commutative Algebra (Fairfax, VA, 1979), Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 201–231.
• S. Huckaba and T. Marley, Hilbert coefficients and the depths of associated graded rings, J. Lond. Math. Soc. (2) 56 (1997), 64–76.
• C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), 293–318.
• E. Hyry, Coefficient ideals and the Cohen–Macaulay property of Rees algebras, Proc. Amer. Math. Soc. 129 (2001), 1299–1308.
• E. Hyry and T. Järvilehto, Hilbert coefficients and the Gorenstein property of the associated graded ring, J. Algebra 273 (2004), 252–273.
• E. Hyry and K. Smith, On a non-vanishing conjecture of Kawamata and the core of an ideal, Amer. J. Math. 125 (2003), 1349–1410.
• A. Kohlhaas, Symmetry in the core of a zero-dimensional monomial ideal, to appear in J. Algebra.
• J. Lipman and A. Sathaye, Jacobian ideals and a theorem of Briançon–Skoda, Michigan Math. J. 28 (1981), 199–222.
• E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer, New York, 2005.
• S. Ohnishi and K. Watanabe, Coefficient ideal of ideals generated by monomials, Comm. Algebra 39 (2011), no. 5, 1563–1576.
• C. Polini and B. Ulrich, A formula for the core of an ideal, Math. Ann. 331 (2005), 487–503.
• C. Polini, B. Ulrich and M. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), 72–93.
• K. Shah, Coefficient ideals, Trans. Amer. Math. Soc. 327 (1991), no. 1, 373–384.
• B. Ulrich, Ideals having the expected reduction number, Amer. J. Math. 118 (1996), 17–38.
• P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101.
• W. Vasconcelos, Integral closure: Rees algebras, multiplicities, algorithms, Springer, Berlin, 2005.