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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.

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
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References:

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