Product Decomposition of non reduced Space Germs

Abstract

It is easy to see, that any N-differentiable space germ X ([12]) can be decomposed as a product X ≃ X₁ x@@@x X_r of space germs X_i (so calledp(roduct)-decomposition [15]) such that no X_i can be decomposed further on (the decomposition and the X_i are then called p-irreducible [15]). The uniqueness of such decompositions is much more difficult to be proved and in fact not completely established up to now. In 1972–78 R. Ephraim succeeded over serveral papers [4]-[7] to establish the uniqueness of p-irreducible pdecompositions for reduced complex analytic space germs. His proof uses havily deep results (starting with coherence) of local complex analytic geometry. C. Becker gave / announced in [2] the same result for coherent real analytic space germs. About 1976 I was able to prove by methods of differentiable geometry the uniqueness for (quite) arbitrary reduced N-differentiable space germs with Ne ∞ ω co (real analytic), ω* (complex analytic), meanwhile also for N ∈ ℕ ([17]). I need only some mild geometric assumption such as “curve rich” [15], [16] for the space germs. This assumption is for example satisfied for each complex-, real- or semi-analytic space germ (curve selection lemma). The differentiable geometric proof of the uniqueness appeared in [15], being rejected earlier because of “the authors own methods”.