THE PREPARATION THEOREM FOR COMPOSITE FUNCTIONS Rm ...
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THE PREPARATION THEOREM FOR COMPOSITE FUNCTIONS Rm ...
by Haynes R. Miller). ABSTRACT. We present a simple .... Hubert maps, such that £p(m) and £â(n) are exactly the sets of G-invariant germs. ¿G(ra) and £o(n), ...
proceedings of the american mathematical Volume
101, Number
society
3, November
1987
THE PREPARATION THEOREM FOR COMPOSITE FUNCTIONS ADAM KOWALCZYK (Communicated
by Haynes R. Miller)
ABSTRACT. We present a simple extension of the preparation Malgrange and J. Mather to the case of composite functions. we obtain a short proof of the equivariant preparation theorem
theorem of B. As a corollary of V. Poénaru.
1. Formulation of the results. We denote by £ (p) the ring of germs of G°°functions at 0 G Rp, by £(p,q) the set of germs at 0 G Rp of G°°-transformations Rp —» R9 which preserve the origin, and by m(p) the maximal ideal of £(p) formed by all functions vanishing at 0. A transformation H E £(p,q) induces a IT*
local ring homomorphism For p E £(m,k)
£(q)—>£(p)
and n G £(n,l)
defined by /3 i—>• /? o H. we introduce
£p(m)
=
{a o p;a
E £(k)},
¿v(n) — {ßorl',ß S £(l)}, mp(m) = m(m)ri(fp(m), and m^n) = m(ri)ni,(fi). Obviously m,,(n) = n*m(l) and mp(m) = p*m(k). A germ / G £(m,n) such that f*£„(n) C £p(m) will be called a prpgerm; for such a transformation each component oinof belongs to £p(m) and so is of the form Fi o p. Hence there exists F E £(k,l) such that the following diagram commutes:
Rk
-^
p
(1)
R v
Rm M
R"
As p*£(k) and f*£n(n) = (nof)*£(l) = (Fop)*£(l) are all subrings of £p(m), any £p(m)-modu\e A could be considered as a £(k)-, £p(n)- or £(/)-module. Obviously ai,...,av generate A as an
