Abstract
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 \(\mathfrak {m}\)-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.
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Notes
From a constructive point of view we cannot deduce that \(\overline{\mathbf {B}}\) is finitely presented from the fact it is finitely generated.
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Supported by Spanish GR MTM-2011-22435 and MTM-2014-55565.
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Alonso, M.E., Lombardi, H. Local Bézout theorem for Henselian rings. Collect. Math. 68, 419–432 (2017). https://doi.org/10.1007/s13348-016-0184-0
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DOI: https://doi.org/10.1007/s13348-016-0184-0