##
**Adjoints of ideals.**
*(English)*
Zbl 1180.13005

Let \(R\) be a regular domain and \(I\) an ideal in \(R\). The adjoint \(\mathrm{adj }I\) of \(I\) is defined by Lipman as
\[
\mathrm{adj } I=\cap_v\{r\in R\mid v(r)\geq v(I)-v(J_{R_v}/R)\},
\]
where the intersection varies over all valuations \(v\) on the field of fractions of \(R\) that are nonnegative on \(R\) and for which the corresponding valuation ring \(R_v\) is a localization of a finitely generated \(R\)-algebra and where \(J_{R_v/R}\) denotes the Jacobian ideal of \(R_v\), over \(R\). The paper under review investigates some properties of adjoints of ideals, in particular for generalized monomial ideals. A crucial property is the subadditivity of adjoints: \(\mathrm{adj}(IJ)\subseteq\mathrm{adj}(I)\mathrm{adj}(J)\). This was proved by different authors, in characteristic \(0\) by J.-P. Demailly, L. Ein and R. Lazarsfeld [Mich. Math. J. 48, Spec. Vol., 137–156 (2000; Zbl 1077.14516)] and for generalized test ideals in characteristic \(p\) by N. Hara and Ken-ichi Yoshida [Trans. Am. Math. Soc. 355, No.8, 3143–3174 (2003; Zbl 1028.13003)]. In the present work, the authors prove the subadditivity of adjoints for generalized monomial ideals and for ideals in two-dimensional regular domains. As a counterexample, the authors give an example of a \(d\)-dimensional regular local ring \((R,M)\), with \(d>2\), which shows that Rees valuations do not suffice in general to define the adjoint of an ideal. Let \(P\) be a prime ideal of \(R\) of height \(h\in\{2,\ldots,d-1\}\) generated by a regular sequence. Then the \(P\)-adic valuation \(v_P\) is the only Rees valuation of \(P\) and \(v_P\) does not define \(\mathrm{adj }(P^{h-1})\). In order to prove the subadditivity of adjoints, it is shown that the Rees valuations suffice for these two types of ideals. In particular,
\[
\mathrm{adj }(I^n)=\cap_v\{r\in R\mid v(r)\geq v(I^n)-v(J_{R_v}/R)\},
\]
where the intersection varies over all Rees valuations \(v\) of \(I\). As a consequence, the adjoint of a (general) monomial ideal is monomial. A similar proof shows that the integral closure of a (general) monomial ideal is also monomial.

Reviewer: Martine Picavet-L’Hermitte (Le Cendre)

### MSC:

13A15 | Ideals and multiplicative ideal theory in commutative rings |

13A18 | Valuations and their generalizations for commutative rings |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

13B22 | Integral closure of commutative rings and ideals |

13F30 | Valuation rings |

### Keywords:

adjoints of ideals; regular domain; generalized monomial ideals; Rees valuations; subadditivity; integral closure of ideals
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\textit{R. Hübl} and \textit{I. Swanson}, Mich. Math. J. 57, 447--462 (2008; Zbl 1180.13005)

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