Doctoral Thesis
DOI
Document
Author
Full name
Euripides Alves da Silva
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 1982
Supervisor
Committee
Favaro, Luiz Antonio (President)
Carneiro, Mario Jorge Dias
Qualifik, Paul
Teixeira, Marco Antonio
Title in Portuguese
CLASSIFICAÇÃO DE PARES BI-ESTÁVEIS POR R-ÁLGEBRAS
Keywords in Portuguese
Não disponível
Abstract in Portuguese
Não disponível
Title in English
Not available
Keywords in English
Not available
Abstract in English
Let f:Rn, 0 → Rp a C map-germ and let us consider the local algebra of order k, QΓ (f) = En / f * Mp + Mk+1n associated with germ f, where En is the ring of germs g : Rn , 0 → R and Mn, is the maximal ideal of germs g : Rn, 0 → R, 0. The Classification ot Stable Germs Theorem through the local algebras is classic: "If f and g are stable, them f and g are A-equivalent if, and only if, the associated algebras are isomorphic"; see, J. Mather [10]. In [3], J.P. Dufour has introduced the notion of stabliitv for couples of germs (f1, f2) : Rn, 0 → Rp x Rq, 0 and has studied the problem of deformations and classification in particular cases, with his own techniques of dlfficult generalization. The objective of this work is the classification of couples of bi-stable germs, by means of the local algebras associated with (f1, f2) and and their components, To reach this objective we introduced the notion of cohorent inomorphiom as follows: Let Φ1 : En / If1 + Mk+1n → En / Ig1 + Mk+1n and Φ2 : En / If2 + Mk+1n → En / Ig2 + Mk+1n, be isomorphisms between two algebras associated with the components of the couples (f1, f2, (g1, g2) : Rn, 0 → Rp x Rq, 0. Let us suppose that there are isomorphism θ1 and θ2 of En, for which we have Φ1 (α + If1 + Mk+1n) = θ1 (α) + Ig1 + Mk+1n and Φ2 (α + If2 + Mk+1n) = Φ2 (α) + I,sub>g2 + Mk+1n. We say that isomorphism Φ1 and Φ2 are induced by Phi;1 and Phi;2, respectivaly. (We observe that whenever f K~g then the algebra Qk(f) and Qk(g) are isomorphic vie an induced isomorphical). We say, then, that the isomorphism Φ1 and Φ2 are coherent when they are indiced by the same isomorphism θ : En → En. (We prove that whenever (f1, f2) Bi-K ~(g1, g2 then the algebras Qk(f1) and Qk(g1, Qk (f2) and Qk(g2 are isomorphic according to coehent isomorphism, i.e., isomorhism induced by the only ring-isumorphisms θ : En → En (see chapter IV, 3). Thus the principal theorem can be enunciated: "If bthe couple of germs (F1, f2) and (g1, g2) are bi-stable, then they are bi-A-equivalent if, and only if, the associated algebras are isomorphic through coherent isomorphisms".