Non-reduced automorphism schemes
type: Item
instance of: scholarly article
title: Non-reduced automorphism schemes
zbMATH Open document ID: 1057.14056
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: Let \(A\) be a finite-dimensional associative algebra over an algebraically closed field \(k\). Denote by \(\mathfrak{Aut}(A)\) the affine group scheme of automorphisms of \(A\) and by \(\mathfrak{Inn}(A)\) the smooth characteristic subgroup of inner automorphisms. Then \(\mathfrak{Out}(A)=\mathfrak{Aut}(A)/\mathfrak{Inn}(A)\) is the affine group scheme of outer automorphisms. By the theorem of Cartier, \(\mathfrak{Aut}(A)\) is always reduced if \(\text{char}(k)=0\), but this is not the case if \(\text{char}(k)=p>0\). The authors prove that the group schemes \(\mathfrak{Aut}(A)\) and \(\mathfrak{Out}(A)\) are not reduced if \(A\) admits a \(\mathbb{Z}_p\)-grading which can not be lifted to a \(\mathbb{Z}\)-grading. The proof is based on the following result: an infinitesimal multiplicative subgroup \(\mathfrak{M}\) of a smooth algebraic group \(\mathfrak{G}\) over an algebraically closed field \(k\) of characteristic \(p>0\) is contained in a maximal torus of \(\mathfrak{G}\). This applies in particular when \(A\) is the group algebra of a \(p\)-group. Another example, where the above result shows that \(\mathfrak{Aut}(A)\) is not reduced, is illustrated by the direct calculation of the automorphism group.
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 2092908
zbMATH Keywords: finite-dimensional associative algebras
zbMATH Keywords: affine group schemes
zbMATH Keywords: automorphism groups
zbMATH Keywords: inner automorphisms
author: Christof Geiss
author: Detlef Voigt
reviewed by: Ivan V. Arzhantsev
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.03.004
OpenAlex ID: W1974613664
cites work: Smooth automorphism group schemes
cites work: An interesting family of algebras
cites work: Geometry of chain complexes and outer automorphisms under derived equivalence.
cites work: Linear algebraic groups.
cites work: Induzierte Darstellungen in der Theorie der endlichen, algebraischen Gruppen
DOI: 10.1016/J.JPAA.2004.03.004
description: scientific article; zbMATH DE number 2092908