Relations among analytic functions. I

Abstract

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).