Group schemes and rigidity of algebras in positive characteristic
type: Item
instance of: scholarly article
title: Group schemes and rigidity of algebras in positive characteristic
zbMATH Open document ID: 0868.17016
DOI: 10.1016/0022-4049(95)00146-8
author: Serge Skryabin
published in: Journal of Pure and Applied Algebra
publication date: 1997-08-21T00:00:00Z
review text: It is well known that in positive (or finite) characteristic additional problems arise in the comparison of the infinitesimal and local points of view. For example trying to ``exponentiate'' the derivation \({\mathcal D}\) of a Lie algebra \(L\) to a formal automorphism \(\varphi=1+t{\mathcal D}+t^2\varphi_2+\cdots\), you meet a problem because the usual formula \(\varphi_n= \frac{1}{n!} {\mathcal D}^n\) stops working for \(n=p\), where \(p\) is the characteristic of the ground field. One can make a cocycle for \(H^2(L,L)\): \[ \psi(x,y)= \sum_{i=1,\dots,p-1} \frac{1}{i!(p-i)!} [{\mathcal D}^ix,{\mathcal D}^{p-i}y], \] which is denoted \(\text{Sq }{\mathcal D}\) and called obstruction. The author makes careful considerations of these problems and of the whole situation with deformation in this context. Following Gerstenhaber, he distinguishes geometric rigidity and formal analytic rigidity, provides several criteria for a finite-dimensional Lie or associative algebra \(L\) to be rigid in that sense, and discusses properties of the obstruction subspace in \(H^2(L,L)\). His Theorem 2 shows the special role of the automorphism scheme \(\Aut(L)\) in this context. As an application the scheme theoretic description of deformations of the Jacobson-Witt algebras \(W_n\) is given.
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 14D15
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 16S80
zbMATH DE Number: 858617
zbMATH Keywords: deformations
zbMATH Keywords: obstructions
zbMATH Keywords: Lie algebra
zbMATH Keywords: group scheme
zbMATH Keywords: geometric rigidity
zbMATH Keywords: formal analytic rigidity
zbMATH Keywords: automorphism scheme
zbMATH Keywords: deformations of the Jacobson-Witt algebras
reviewed by: Alexei Rudakov
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00146-8
OpenAlex ID: W2040575278
cites work: A theory of linear descent based upon Hopf algebraic techniques
cites work: Q5602064
cites work: Q5751032
cites work: On the deformation of rings and algebras
cites work: On the deformation of rings and algebras. III
cites work: Relative Hochschild cohomology, rigid algebras, and the Bockstein
cites work: On the Deformation of Algebra Morphisms and Diagrams
cites work: Commutative Algebras and Cohomology
cites work: Q4143433
cites work: The red book of varieties and schemes
cites work: Q5551287
cites work: Commutative algebra cohology and deformations of Lie and associative algebras
cites work: Effacement et déformation
cites work: Deformation of transitive Lie algebras
cites work: Q5638468
cites work: Automorphism schemes and forms of Witt Lie algebras
description: scientific article; zbMATH DE number 858617