Local Bézout theorem for Henselian rings

• Autores: María Emilia Alonso García , Henri Lombardi
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 3, 2017, págs. 419-432
• Idioma: inglés
• DOI: 10.1007/s13348-016-0184-0
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• Resumen

  • 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.