Doctoral Thesis

DOI

https://doi.org/10.11606/T.55.2019.tde-21112019-154256

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

Tadini, Wilson Mauricio

Teixeira, Marco Antonio

Carneiro, Mario Jorge Dias

Qualifik, Paul

Tadini, Wilson Mauricio

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:R^{n}, 0 → R^{p} a C^{∞} map-germ and let us consider the local algebra of order k, Q_{Γ} (f) = E_{n} / f * M_{p} + M^{k+1}_{n} associated with germ f, where E_{n} is the ring of germs g : R^{n} , 0 → R and M_{n}, is the maximal ideal of germs g : R^{n}, 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 (f_{1}, f_{2}) : R^{n}, 0 → R^{p} x R^{q}, 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 (f_{1}, f_{2}) and and their components, To reach this objective we introduced the notion of cohorent inomorphiom as follows: Let Φ_{1} : E_{n} / I_{f}_{1} + M^{k+1}_{n} → E_{n} / I_{g}_{1} + M^{k+1}_{n} and Φ_{2} : E_{n} / I_{f}_{2} + M^{k+1}_{n} → E_{n} / I_{g}_{2} + M^{k+1}_{n}, be isomorphisms between two algebras associated with the components of the couples (f_{1}, f_{2}, (g_{1}, g_{2}) : R^{n}, 0 → R^{p} x R^{q}, 0. Let us suppose that there are isomorphism θ_{1 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>g}_{2} + M^{k+1}_{n}. We say that isomorphism Φ_{1} and Φ_{2} are induced by Phi;_{1} and Phi;_{2}, respectivaly. (We observe that whenever f K~g then the algebra Q_{k}(f) and Q_{k}(g) are isomorphic vie an induced isomorphical). We say, then, that the isomorphism Φ_{1} and Φ_{2} are coherent when they are indiced by the same isomorphism θ : E_{n} → E_{n}. (We prove that whenever (f_{1}, f_{2}) Bi-K ~(g_{1}, g_{2} then the algebras Q_{k}(f_{1}) and Q_{k}(g_{1}, Q_{k} (f_{2}) and Q_{k}(g_{2} are isomorphic according to coehent isomorphism, i.e., isomorhism induced by the only ring-isumorphisms θ : E_{n} → E_{n} (see chapter IV, 3). Thus the principal theorem can be enunciated: "If bthe couple of germs (F_{1}, f_{2}) and (g_{1}, g_{2}) are bi-stable, then they are bi-A-equivalent if, and only if, the associated algebras are isomorphic through coherent isomorphisms".

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EuripedesAlvesdaSilva_DO.pdf (4.53 Mbytes)

Publishing Date

2019-11-26

CeTI-SC/STI

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