Varieties and modules of small dimension
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.
scientific article; zbMATH DE number 446369
scientific article