Relations among analytic functions. I

Bierstone, Edward; Milman, P. D.

Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 187-239.

Ni les ensembles analytiques réels, ni les images d’applications analytiques réelles ou complexes sont cohérents, en général. Soit $\Phi :X\to Y$ un morphisme d’espaces analytiques, et soit $\Psi :𝒢\to ℱ$ un homomorphisme de modules cohérents au-dessus de l’homomorphisme induit d’anneaux ${\Phi }^{*}:{𝒪}_Y\to {𝒪}_X$. On conjecture que, malgré le manque de cohérence, certains invariants discrets naturels des modules de relations formelles ${ℛ}_a=\mathrm{Ker}{\widehat{\Psi }}_a$, $a\in X$, sont semicontinus supérieurement pour la topologie de Zariski analytique de $X$. On démontre la semicontinuité dans plusieurs cas (par exemple, dans la catégorie algébrique). La semicontinuité du “diagramme des exposants initiaux” fournit un point de vue unifié et des techniques nouvelles et explicites qui se substituent à la cohérence dans des problèmes géoémtriques sur les images d’applications (ensembles semianalytiques ou sousanalytiques) et dans des problèmes analytiques sur les singularités de fonctions différentiables (en particulier, les problèmes classiques de division et composition).

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\to Y$ be a morphism of real analytic spaces, and let $\Psi :𝒢\to ℱ$ be a homomorphism of coherent modules over the induced ring homomorphism ${\Phi }^{*}:{𝒪}_Y\to {𝒪}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ${ℛ}_a=\mathrm{Ker}{\widehat{\Psi }}_a$, $a\in X$, are upper semi-continuous in the analytic Zariski topology of $X$. We prove semicontinuity in many cases (e.g. in the algebraic category). Semicontinuity of the “diagram of initial exponents” provides a unified point of view and explicit new techniques which substitute for coherence in both geometric problems on the images of mappings (semianalytic and subanalytic sets) and analytic problems on the singularities of differentiable functions (in particular, the classical division and composition problems).

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