Relations among analytic functions. I

Edward Bierstone; P. D. Milman

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 1, page 187-239
  • ISSN: 0373-0956

Abstract

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Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ : X → Y be a morphism of real analytic spaces, and let Ψ : 𝒢 → ℱ be a homomorphism of coherent modules over the induced ring homomorphism Φ * : 𝒪 Y → 𝒪 X . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ℛ a = Ker Ψ ^ a , a ∈ 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).

How to cite

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Bierstone, Edward, and Milman, P. D.. "Relations among analytic functions. I." Annales de l'institut Fourier 37.1 (1987): 187-239. <http://eudml.org/doc/74743>.

@article{Bierstone1987,
abstract = {Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\rightarrow Y$ be a morphism of real analytic spaces, and let $\Psi :\{\cal G\}\rightarrow \{\cal F\}$ be a homomorphism of coherent modules over the induced ring homomorphism $\Phi ^*:\{\cal O\}_Y\rightarrow \{\cal O\}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations $\{\cal R\}_a=~\{\rm Ker\}\widehat\{\Psi \}_a$, $a\in X$, 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).},
author = {Bierstone, Edward, Milman, P. D.},
journal = {Annales de l'institut Fourier},
keywords = {real analytic spaces; semicontinuity; singularities of differentiable functions; division; composition; analytic function},
language = {eng},
number = {1},
pages = {187-239},
publisher = {Association des Annales de l'Institut Fourier},
title = {Relations among analytic functions. I},
url = {http://eudml.org/doc/74743},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Bierstone, Edward
AU - Milman, P. D.
TI - Relations among analytic functions. I
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 1
SP - 187
EP - 239
AB - Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\rightarrow Y$ be a morphism of real analytic spaces, and let $\Psi :{\cal G}\rightarrow {\cal F}$ be a homomorphism of coherent modules over the induced ring homomorphism $\Phi ^*:{\cal O}_Y\rightarrow {\cal O}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ${\cal R}_a=~{\rm Ker}\widehat{\Psi }_a$, $a\in X$, 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).
LA - eng
KW - real analytic spaces; semicontinuity; singularities of differentiable functions; division; composition; analytic function
UR - http://eudml.org/doc/74743
ER -

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