The group of automorphisms of a commutative algebra
type: Item
instance of: scholarly article
title: The group of automorphisms of a commutative algebra
zbMATH Open document ID: 0824.16025
DOI: 10.1007/BF02572348
author: Francisco Guil-Asensio
author: Manuel Saorín
published in: Mathematische Zeitschrift
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/174756
review text: We study the automorphism group \(\text{Aut}(A)\) of a local commutative algebra \(A\) and its identity component \(G_ A\). The monomial algebras are then characterized as those for which \(G_ A\) has rank \(n = \dim J/J^ 2\) and, in particular, the algebras of the form \(K[X_ 1,\dots, X_ n]/ (X_ 1, \dots, X_ n)^ \ell\) are characterized as those for which \(\text{Aut} (A)\) is a semidirect product of a unipotent normal subgroup and the linear group \(\text{GL}(n,K)\). A general strategy to study \(\text{Aut} (A)\) via \(\text{Aut} (G(A))\), where \(G(A)\) is the associated graded algebra, is also developed. That strategy allows us to prove that, when \(n = 2\) and \(A\) is not isomorphic to \(K[X,Y] / (X,Y)^ \ell\), \(G_ A\) is a solvable group. If \(U_ A\) is the (unipotent) kernel of the canonical morphism \(\text{Aut} (A) \to \text{GL} (J/J^ 2)\), then a precise description of the quotient \(G_ A/U_ A\) is given, when \(A\) is gradable by the radical. In the final section it is seen that, again with \(n = 2\), no algebra with Loewy length \(L(A) \leq 4\) has unipotent \(G_ A\) and we identify, up to isomorphism, all those with \(L(A) = 5\) for which \(G_ A\) is unipotent or \(\text{Aut}(A)\) is nilpotent.
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 20F29
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 784098
zbMATH Keywords: automorphism group
zbMATH Keywords: local commutative algebra
zbMATH Keywords: semidirect product
zbMATH Keywords: associated graded algebra
zbMATH Keywords: solvable group
zbMATH Keywords: Loewy length
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784098