Varieties and modules of small dimension
type: Item
instance of: scholarly article
title: Varieties and modules of small dimension
zbMATH Open document ID: 0783.20004
DOI: 10.1007/BF01202306
published in: Archiv der Mathematik
publication date: 1993-11-15T00:00:00Z
review text: Let \(E\) be an elementary abelian group of order \(p^ n\) and let \(k\) be a field of characteristic \(p\). Suppose that \(M\) is a \(kE\)-module of complexity \(n-t\) and dimension \(dp^ t\). Then the dimension of its variety is \(n-t\), and some elementary intersection theory implies that the variety of \(M\) must have degree at most \(d\). In the linear case, this observation permits us to give a very swift and direct proof of a recent result of Alperin and Mason. Their theorem says that if \(G=SL(2,p^ n)\), \(n>1\), and if \(M\) is \(kG\)-module such that the restriction of \(M\) to some elementary abelian \(p\)-subgroup \(E\) of \(G\) is isomorphic to \(kE\), then \(M\) is a simple module. In fact the theorem of Alperin and Mason is slightly generalized in this paper.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 446369
zbMATH Keywords: elementary abelian group
zbMATH Keywords: complexity
zbMATH Keywords: dimension
zbMATH Keywords: variety
zbMATH Keywords: intersection theory
zbMATH Keywords: simple module
author: Jon F. Carlson
MaRDI profile type: Publication
cites work: On Simple Modules for SL(2, q)
cites work: Q3997902
cites work: The varieties and the cohomology ring of a module
cites work: The Cohomology of Irreducible Modules Over SL(2,pn )
cites work: Wild categories of periodic modules
cites work: Q4001605
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description: scientific article; zbMATH DE number 446369