Complexity and varieties for infinite groups. I (Q1364267)

The author develops the module theory and the theory of complexity and varieties for modules of type \(FP_\infty\) over \(LH{\mathfrak F}\)-groups, which have been introduced by P. H. Kropholler. As applications of these methods the author obtains the following results. Theorem 1.1. Let \(G\) be an \(LH{\mathfrak F}\)-group, \(k\) a commutative Noetherian ring of finite global dimension, \(M\) a \(kG\)-module of type \(FP_\infty\). Every element of \(\text{Ext}^*_{kG}(M,M)\), which restricts to a nilpotent element of each finite elementary abelian subgroup of \(G\), is nilpotent. Theorem 1.2. Let \(G\) be an \(LH{\mathfrak F}\)-group, \(k\) a field, \(M\) a \(kG\)-module of type \(FP_\infty\). Then there is an upper bound for the complexities of the restrictions of \(M\) to finite subgroups of \(G\).