##
**Muhly local domains and Zariski’s theory of complete ideals.**
*(English)*
Zbl 1376.13004

The paper under review is concerned with certain properties of integrally closed ideals in two-dimensional normal local domains \((R,\mathfrak{m})\) with algebraically closed residue field and with the associated graded ring \(\text{gr}_{\mathfrak m}R\) an integrally closed domain. The author refers to such local domains as two-dimensional Muhly local domains. (Note that a two-dimensional Muhly local domain is Cohen-Macaulay because of being normal.)

The study of integrally closed ideals started by Zariski in two-dimensional regular local rings, in [O. Zariski, Am. J. Math. 60, 151–204 (1938; Zbl 0018.20101)]. (Also see Appendix 5 in [O. Zariski and P. Samuel, Commutative Algebra Vol. II. Reprint of the 1958-1960 Van Nostrand edition. Graduate Texts in Mathematics. 29. New York-Heidelberg-Berlin: Springer-Verlag (1976; Zbl 0322.13001)].) According to Chapter 14 in [I. Swanson and C. Huneke, Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series 336. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)]: “Zariski’s motivation was to give algebraic meaning to the idea of complete linear systems of curves defined by base conditions in which the curves pass through prescribed base points with given multiplicities.” In the early 1960s, in a series of papers: [H. T. Muhly and M. Sakuma, J. Lond. Math. Soc. 38, 341–350 (1963; Zbl 0142.28802)], [H. T. Muhly and M. Sakuma, Trans. Am. Math. Soc. 106, 210–221 (1963; Zbl 0123.03601)], and [H. T. Muhly, J. Lond. Math. Soc. 40, 99–107 (1965; Zbl 0123.03701)] Muhly attempted to extend some of Zariski’s results to (and this is the reason for the terminology) two-dimensional Muhly local domains.

We recall a few definitions before stating the main results of the paper under review:

(a) An ideal \(I\) of a Noetherian local ring is called simple if \(I\neq J\cdot K\) for any proper ideals \(J\) and \(K\). An element \(x\in R\) is said to be integral over an ideal \(I\) if \(x\) satisfies an equation of the form \(x^n+r_1x^{n-1}+\cdots+r_n=0\), where \(r_i\in I^i\). The set of all elements in \(R\) that are integral over an ideal \(I\) forms an ideal, denoted by \(\overline{I}\) and called the integral closure of \(I\). An ideal \(I\) is said to be integrally closed (or complete) if \(I=\overline{I}\).

(b) In a Noetherian local integral domain \((R,\mathfrak{m})\), a prime divisor \(v\) of \(R\) is a discrete valuation \(v\) of the quotient field of \(R\), with value group \(\mathbb{Z}\), whose valuation ring \((V,\mathfrak{m}_V)\) dominates \((R,\mathfrak{m})\) and such that the transcendence degree \(\text{tr\;deg}_{R/\mathfrak{m}}V/\mathfrak{m}_V=\dim R-1\). (Cf. [D. Rees and R. Y. Sharp, J. Lond. Math. Soc., II. Ser. 18, 449–463 (1978; Zbl 0408.13009)].)

(c) If \(R\) is a Noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(x\in\mathfrak{a}\), then \(R[\mathfrak{a}/x]\) denotes the subring of \(R_x\) generated by the image of \(R\) and by the set \(\{a/x\mid a\in\mathfrak{a}\}\).

(d) In a two-dimensional Muhly local domain \((R,\mathfrak{m})\), an \(\mathfrak{m}\)-primary ideal \(I\) is said to be contracted from the blowup of \(R\) at \(\mathfrak{m}\), \(\text{Bl}_{\mathfrak{m}}R\), if \[ I=\bigcap_{S\in\Sigma} IS\cap R, \] where \[ \Sigma=\left\{R\left[\frac{\mathfrak{m}}{x}\right]_{\mathfrak p}\mid 0\neq x\in\mathfrak{m}, \mathfrak{p}\in\text{Spec}\left(R\left[\frac{\mathfrak{m}}{x}\right]\right)\right\}. \] For example, it can be shown that any complete \(\mathfrak{m}\)-primary ideal of \((R,\mathfrak{m})\) is contracted from \(\text{Bl}_{\mathfrak{m}}R\). Note that \(\Sigma\) in the above definition coincides with the set of local rings of the scheme \(\text{Bl}_{\mathfrak{m}}R\).

(e) Let \((R,\mathfrak{m})\) be a two-dimensional Muhly local domain, and let \[ R\left[\frac{\mathfrak{m}}{x}\right]_{\mathfrak n} \] be a (local) quadratic transformation of \(R\) (that is, \(x\in\mathfrak{m}\setminus\mathfrak{m}^2\) and \(\mathfrak{n}\) a maximal ideal of \(R[\;mathfrak{m}/x]\) such that \(\mathfrak{n}\cap R=\mathfrak{m}\)). Let \(I\) be an \(\mathfrak{m}\)-primary ideal of \(R\) with \(\text{ord}_R(I)=r\), where \(\text{ord}_R(I)=\max\{k\mid I\subset\mathfrak{m}^k\}\). Then it follows that in \(R[\mathfrak{m}/x]\): \[ IR\left[\frac{\mathfrak{m}}{x}\right]=x^r J, \] with \(J\) an ideal of \(R[\mathfrak{m}/x]\). This ideal \(J\) is called the transform of \(I\) in \(R[\mathfrak{m}/x]\).

Zariski proved in [loc. cit.] that in a two-dimensional regular local ring \((R,\mathfrak{m})\):

(1) The product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted.

(2) The product of integrally closed ideals is integrally closed.

(3) The transform of a simple (\(\mathfrak{m}\)-primary) integrally closed ideal is a simple integrally closed ideal.

(4) Every \(\mathfrak{m}\)-primary integrally closed ideal factors uniquely, up to order, as a finite product of simple integrally closed ideals.

(5) There exists a one-to-one correspondence between the set of simple (\(\mathfrak{m}\)-primary) integrally closed ideals and the set of prime divisors.

In a two-dimensional Cohen-Macaulay local domain \(R\), by Abhyankar’s inequality we have: \(\text{emb\;dim}R\leq e(R)+1\). (See [S. S. Abhyankar, Am. J. Math. 89, 1073–1077 (1967; Zbl 0159.33202)].) If equality holds then \(R\) is said to have minimal multiplicity. One of the results of the paper under review is that a two-dimensional Muhly local domain \(R\) satisfies the property that any product of integrally closed ideals is integrally closed ((2) from Zariski’s results listed above) if and only if \(R\) has minimal multiplicity. Moreover, if a two-dimensional Muhly local domain \((R,\mathfrak{m})\) has minimal multiplicity, then \(R\) satisfies the property that any product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted ((1) from Zariski’s results listed above). The converse of this statement is shown to hold under an additional hypothesis, called condition (MS) in the paper. Condition (MS) is a hypothesis concerning the associated graded ring \(\text{gr}_{\mathfrak m}R\) of a a two-dimensional Muhly local domain \((R,\mathfrak{m})\) and was assumed by Muhly and Sakuma in [loc. cit.]. Condition (MS) reads: “If \(\alpha\) and \(\beta\) are two homogeneous elements of \(\text{gr}_{\mathfrak m}R\) of degree \(r\) and \(s\) respectively such that the ideal \((\alpha,\beta)\text{gr}_{\mathfrak m}R\) is irrelevant, then the ideal \((\alpha,\beta)\text{gr}_{\mathfrak m}R\) contains all homogeneous elements of \(\text{gr}_{\mathfrak m}R\) of degree not less than \(r + s\).” Thus, it is shown that if a two-dimensional Muhly local domain \(R\) satisfies condition (MS), then \(R\) satisfies the property that any product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted ((1) from Zariski’s results listed above) if and only if \(R\) has minimal multiplicity.

The author shows, by means of counter-examples that the remaining results (3)–(5) of Zariski from the above list do not necessarily hold in a two-dimensional Muhly local domain \(R\), even if \(R\) has minimal multiplicity. There is also an example that shows if a two-dimensional Muhly local domain \(R\) does not have minimal multiplicity, then the property“any product of contracted ideals is contracted” may not necessarily hold in \(R\). Finally, the author shows, by means of another example that there exist two-dimensional Muhly local domains that satisfy condition (MS) but do not have minimal multiplicity.

The study of integrally closed ideals started by Zariski in two-dimensional regular local rings, in [O. Zariski, Am. J. Math. 60, 151–204 (1938; Zbl 0018.20101)]. (Also see Appendix 5 in [O. Zariski and P. Samuel, Commutative Algebra Vol. II. Reprint of the 1958-1960 Van Nostrand edition. Graduate Texts in Mathematics. 29. New York-Heidelberg-Berlin: Springer-Verlag (1976; Zbl 0322.13001)].) According to Chapter 14 in [I. Swanson and C. Huneke, Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series 336. Cambridge: Cambridge University Press (2006; Zbl 1117.13001)]: “Zariski’s motivation was to give algebraic meaning to the idea of complete linear systems of curves defined by base conditions in which the curves pass through prescribed base points with given multiplicities.” In the early 1960s, in a series of papers: [H. T. Muhly and M. Sakuma, J. Lond. Math. Soc. 38, 341–350 (1963; Zbl 0142.28802)], [H. T. Muhly and M. Sakuma, Trans. Am. Math. Soc. 106, 210–221 (1963; Zbl 0123.03601)], and [H. T. Muhly, J. Lond. Math. Soc. 40, 99–107 (1965; Zbl 0123.03701)] Muhly attempted to extend some of Zariski’s results to (and this is the reason for the terminology) two-dimensional Muhly local domains.

We recall a few definitions before stating the main results of the paper under review:

(a) An ideal \(I\) of a Noetherian local ring is called simple if \(I\neq J\cdot K\) for any proper ideals \(J\) and \(K\). An element \(x\in R\) is said to be integral over an ideal \(I\) if \(x\) satisfies an equation of the form \(x^n+r_1x^{n-1}+\cdots+r_n=0\), where \(r_i\in I^i\). The set of all elements in \(R\) that are integral over an ideal \(I\) forms an ideal, denoted by \(\overline{I}\) and called the integral closure of \(I\). An ideal \(I\) is said to be integrally closed (or complete) if \(I=\overline{I}\).

(b) In a Noetherian local integral domain \((R,\mathfrak{m})\), a prime divisor \(v\) of \(R\) is a discrete valuation \(v\) of the quotient field of \(R\), with value group \(\mathbb{Z}\), whose valuation ring \((V,\mathfrak{m}_V)\) dominates \((R,\mathfrak{m})\) and such that the transcendence degree \(\text{tr\;deg}_{R/\mathfrak{m}}V/\mathfrak{m}_V=\dim R-1\). (Cf. [D. Rees and R. Y. Sharp, J. Lond. Math. Soc., II. Ser. 18, 449–463 (1978; Zbl 0408.13009)].)

(c) If \(R\) is a Noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(x\in\mathfrak{a}\), then \(R[\mathfrak{a}/x]\) denotes the subring of \(R_x\) generated by the image of \(R\) and by the set \(\{a/x\mid a\in\mathfrak{a}\}\).

(d) In a two-dimensional Muhly local domain \((R,\mathfrak{m})\), an \(\mathfrak{m}\)-primary ideal \(I\) is said to be contracted from the blowup of \(R\) at \(\mathfrak{m}\), \(\text{Bl}_{\mathfrak{m}}R\), if \[ I=\bigcap_{S\in\Sigma} IS\cap R, \] where \[ \Sigma=\left\{R\left[\frac{\mathfrak{m}}{x}\right]_{\mathfrak p}\mid 0\neq x\in\mathfrak{m}, \mathfrak{p}\in\text{Spec}\left(R\left[\frac{\mathfrak{m}}{x}\right]\right)\right\}. \] For example, it can be shown that any complete \(\mathfrak{m}\)-primary ideal of \((R,\mathfrak{m})\) is contracted from \(\text{Bl}_{\mathfrak{m}}R\). Note that \(\Sigma\) in the above definition coincides with the set of local rings of the scheme \(\text{Bl}_{\mathfrak{m}}R\).

(e) Let \((R,\mathfrak{m})\) be a two-dimensional Muhly local domain, and let \[ R\left[\frac{\mathfrak{m}}{x}\right]_{\mathfrak n} \] be a (local) quadratic transformation of \(R\) (that is, \(x\in\mathfrak{m}\setminus\mathfrak{m}^2\) and \(\mathfrak{n}\) a maximal ideal of \(R[\;mathfrak{m}/x]\) such that \(\mathfrak{n}\cap R=\mathfrak{m}\)). Let \(I\) be an \(\mathfrak{m}\)-primary ideal of \(R\) with \(\text{ord}_R(I)=r\), where \(\text{ord}_R(I)=\max\{k\mid I\subset\mathfrak{m}^k\}\). Then it follows that in \(R[\mathfrak{m}/x]\): \[ IR\left[\frac{\mathfrak{m}}{x}\right]=x^r J, \] with \(J\) an ideal of \(R[\mathfrak{m}/x]\). This ideal \(J\) is called the transform of \(I\) in \(R[\mathfrak{m}/x]\).

Zariski proved in [loc. cit.] that in a two-dimensional regular local ring \((R,\mathfrak{m})\):

(1) The product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted.

(2) The product of integrally closed ideals is integrally closed.

(3) The transform of a simple (\(\mathfrak{m}\)-primary) integrally closed ideal is a simple integrally closed ideal.

(4) Every \(\mathfrak{m}\)-primary integrally closed ideal factors uniquely, up to order, as a finite product of simple integrally closed ideals.

(5) There exists a one-to-one correspondence between the set of simple (\(\mathfrak{m}\)-primary) integrally closed ideals and the set of prime divisors.

In a two-dimensional Cohen-Macaulay local domain \(R\), by Abhyankar’s inequality we have: \(\text{emb\;dim}R\leq e(R)+1\). (See [S. S. Abhyankar, Am. J. Math. 89, 1073–1077 (1967; Zbl 0159.33202)].) If equality holds then \(R\) is said to have minimal multiplicity. One of the results of the paper under review is that a two-dimensional Muhly local domain \(R\) satisfies the property that any product of integrally closed ideals is integrally closed ((2) from Zariski’s results listed above) if and only if \(R\) has minimal multiplicity. Moreover, if a two-dimensional Muhly local domain \((R,\mathfrak{m})\) has minimal multiplicity, then \(R\) satisfies the property that any product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted ((1) from Zariski’s results listed above). The converse of this statement is shown to hold under an additional hypothesis, called condition (MS) in the paper. Condition (MS) is a hypothesis concerning the associated graded ring \(\text{gr}_{\mathfrak m}R\) of a a two-dimensional Muhly local domain \((R,\mathfrak{m})\) and was assumed by Muhly and Sakuma in [loc. cit.]. Condition (MS) reads: “If \(\alpha\) and \(\beta\) are two homogeneous elements of \(\text{gr}_{\mathfrak m}R\) of degree \(r\) and \(s\) respectively such that the ideal \((\alpha,\beta)\text{gr}_{\mathfrak m}R\) is irrelevant, then the ideal \((\alpha,\beta)\text{gr}_{\mathfrak m}R\) contains all homogeneous elements of \(\text{gr}_{\mathfrak m}R\) of degree not less than \(r + s\).” Thus, it is shown that if a two-dimensional Muhly local domain \(R\) satisfies condition (MS), then \(R\) satisfies the property that any product of contracted (\(\mathfrak{m}\)-primary) ideals is contracted ((1) from Zariski’s results listed above) if and only if \(R\) has minimal multiplicity.

The author shows, by means of counter-examples that the remaining results (3)–(5) of Zariski from the above list do not necessarily hold in a two-dimensional Muhly local domain \(R\), even if \(R\) has minimal multiplicity. There is also an example that shows if a two-dimensional Muhly local domain \(R\) does not have minimal multiplicity, then the property“any product of contracted ideals is contracted” may not necessarily hold in \(R\). Finally, the author shows, by means of another example that there exist two-dimensional Muhly local domains that satisfy condition (MS) but do not have minimal multiplicity.

Reviewer: Mahdi Majidi-Zolbanin (New York)

### MSC:

13B22 | Integral closure of commutative rings and ideals |

13A35 | Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |

### Keywords:

regular local ring; Muhly local domain; complete ideal; contracted ideal; integrally closed ideal### Citations:

Zbl 0018.20101; Zbl 0322.13001; Zbl 1117.13001; Zbl 0142.28802; Zbl 0123.03601; Zbl 0123.03701; Zbl 0408.13009; Zbl 0159.33202
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\textit{R. Debremaeker}, J. Commut. Algebra 5, No. 4, 507--526 (2013; Zbl 1376.13004)

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### References:

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