Clôture intégrale des idéaux et équisingularité

Lejeune-Jalabert, Monique; Teissier, Bernard

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 4, pp. 781-859.

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.

The first part begins with the definition and elementary properties of the $\overline{\nu }$ order function associated to an ideal $I$ of a reduced analytic algebra $A$. Denoting by ${\nu }_I\left(x\right)$ the largest power of $I$ containing the element $x\in A$, one defines ${\overline{\nu }}_I\left(x\right)={\text{lim}}_{i\to \infty }{\nu }_I\left(x^k\right)/k$. The second paragraph is devoted to the equivalent definitions of the integral closure of an ideal in complex analytic geometry, one of them being $\overline{I}=\{x\in A/{\overline{\nu }}_I\left(x\right)\ge 1\}$. The third paragraph describes the normalized blowing-up of an ideal and the fourth explains how to compute ${\overline{\nu }}_I\left(x\right)$ with the help of the normalized blowing-up of the ideal $I$. It contains the basic finiteness results of the seminar, such as the rationality of ${\overline{\nu }}_I\left(x\right)$ (which had been proved by Nagata in algebraic geometry, a fact of which we were not aware at the time), the definitions of the fractional powers of coherent sheaves of ideals and the proof of their coherency. Given a coherent sheaf $ℐ$ of ${𝒪}_X$-ideals on a reduced analytic space $X$ one can define for each open set $U$ of $X$ and $f\in \Gamma (U,{𝒪}_X)$ the number ${\overline{\nu }}_{ℐ}^U\left(f\right)$ as the infimum of the ${\overline{\nu }}_{{ℐ}_y}\left(f_y\right)$ for $y\in U$.

Then one defines for each positive real number $\nu$ the sheaf $\overline{{ℐ}^{\nu }}$ (resp. $\overline{{ℐ}^{\nu +}}$) associated to the presheaf

$$U\mapsto \{f\in \Gamma (U,{𝒪}_X)/{\overline{\nu }}_{ℐ}^U\left(f\right)\ge \nu \}$$

(resp.

$$U\mapsto \{f\in \Gamma (U,{𝒪}_X)/{\overline{\nu }}_{ℐ}^U\left(f\right)>\nu \}).$$

Finally one has the graded ${𝒪}_X/ℐ$-algebra

$${\overline{\text{gr}}}_{ℐ}{𝒪}_X=\underset{\nu \in {ℝ}_0}{⨁}\overline{{ℐ}^{\nu }}/\overline{{ℐ}^{\nu +}}.$$

One important result is then that this algebra is locally finitely generated and that locally there is a universal denominator $q$ in the sense that all nonzero homogeneous components of the graded algebra have degree in $\frac{1}{q}\mathbb{N}$.

In § 5 it is shown that one can compute $\overline{\nu }$ using analytic arcs $h:(ℂ,0)\to (X,x)$, and § 6 shows that Łojasiewicz exponents are the inverses of $\overline{\nu }$, which implies that they are rational.

Risler’s appendix shows how to use blowing-ups to compute Łojasiewicz exponents and prove their rationality in the real analytic case.

The complements, added for this publication, point to some developments directly related to the subject of the seminar:

The first one is the proof in the spirit of the seminar of the classical Łojasiewicz inequality $\left|\text{grad}\left(f\left(z\right)\right)\right|\ge C_1{\left|f\left(z\right)\right|}^{\theta }$ with $\theta <1$.

Then we point to later work which shows that in fact given an ideal $I$ and an element $f\in A$ the rational number ${\overline{\nu }}_I\left(f\right)$ can be seen as the slope of one of the sides of a natural Newton polygon associated to $I$ and $f$, which is in several ways a better indicator of the relations of the powers of $f$ with the powers of $I$ and has some useful incarnations. The third complement points to results of Izumi using $\overline{\nu }$ to characterize the Gabrielov rank condition for a morphism of analytic algebras, the fourth is a presentation of a generalization due to Ciuperča, Enescu and Spiroff of the rationality of $\overline{\nu }$ to the case of several ideals, where it becomes the rationality of a certain polyhedral cone.

The fifth comment presents the connection of $\overline{\nu }$ with the type of ideals, which was introduced by D’Angelo in complex analysis and used recently by Heier for the proof of an effective Nullstellensatz. In the middle 1980’s, A. Płoski, J. Chadzyński and T. Krasiński found methods of evaluation for the local and global Łojasiewicz exponents in inequalities of the form $\left|P\left(z\right)\right|\ge {C\left|z\right|}^{\theta }$ where either $P=(P_1,...,P_k)$ is a collection of analytic functions on ${ℂ}^n$ having an isolated zero at the origin and the inequality should be true for $\left|z\right|$ small enough, or $P$ is a collection of polynomials with finitely many common zeroes and the inequality should be true for $\left|z\right|$ large enough. The results on the type are of the same nature, because it follows from the seminar that the type is in fact a Łojasiewicz exponent.

The sixth comment points to results of Morales and others about the Hilbert function associated to the integrally closed powers $\overline{I^n}$ of a primary ideal in an excellent local ring and the associated graded algebra.

Finally we point to two different but not unrelated uses of what is in fact the main object of study in the seminar: the reduced graded ring ${\overline{\text{gr}}}_{I}A$ defined and studied in § 4. In [T5] the second author uses the fact that for the local algebra $𝒪$ of a plane analytic branch the algebra ${\overline{\text{gr}}}_m𝒪$ is the algebra of the semigroup associated to the singularity and is a complete intersection (a result due to the first author) to revisit the local moduli problem. The key is that the local analytic algebra $𝒪$ of every plane branch in the same equisingularity class has the same ${\overline{\text{gr}}}_m𝒪$ because it has the same semigroup, so that the branch is a deformation of the monomial curve corresponding to that algebra. In [Kn], Allen Knutson uses the same specialization to the “balanced normal cone" corresponding to ${\overline{\text{gr}}}_{I}A$ in intersection theory.

Each paragraph has its own bibliography. Unfortunately at the time of the seminar we were unaware of the beautiful results of Samuel, Rees and Nagata (see [Sa], [N], [R1], [R2], [R3] in the bibliography of the complements), of which it appears a posteriori that some parts of the seminar are translations into the complex analytic framework. The demand for this text over the years, however, and the fact that some mathematicians are led to rediscover some of its results, indicate that its publication is probably of some use.

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