Local Bézout theorem for Henselian rings

  1. M. Emilia Alonso García
  2. Henri Lombardi
Revista:
Collectanea mathematica

ISSN: 0010-0757

Ano de publicación: 2017

Volume: 68

Fascículo: 3

Páxinas: 419-432

Tipo: Artigo

DOI: 10.1007/S13348-016-0184-0 DIALNET GOOGLE SCHOLAR lock_openAcceso aberto editor

Outras publicacións en: Collectanea mathematica

Resumo

In this paper we prove what we call Local Bézout Theorem (Theorem 3.7). It is a formal abstract algebraic version, in the frame of Henselian rings and � -adic topology, of a well known theorem in the analytic complex case. This classical theorem says that, given an isolated point of multiplicity r as a zero of a local complete intersection, after deforming the coefficients of these equations we find in a sufficiently small neighborhood of this point exactly r isolated zeroes counted with multiplicities. Our main tools are, first the border bases [11], which turned out to be an efficient computational tool to deal with deformations of algebras. Second we use an important result of de Smit and Lenstra [7], for which there exists a constructive proof in [13]. Using these tools we find a very simple proof of our theorem, which seems new in the classical literature.

Información de financiamento

Financiadores

  • Spanish GR
    • MTM-2011-22435
    • MTM-2014-55565
  • Spanish GR
    • MTM-2014-55565
    • MTM-2011-22435