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Relations among analytic functions. I

Edward BierstoneP. D. Milman — 1987

Annales de l'institut Fourier

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 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 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)....

Relations among analytic functions. II

Edward BierstoneP. D. Milman — 1987

Annales de l'institut Fourier

This is a sequel to “Relations among analytic functions I”, , , fasc. 1, [pp. 187-239]. We reduce to semicontinuity of local invariants the problem of finding 𝒞 solutions to systems of equations involving division and composition by analytic functions. We prove semicontinuity in several general cases : in the algebraic category, for “regular” mappings, and for module homomorphisms over a finite mapping.

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