Commutative algebra for cohomology rings of virtual duality groups
\textit{D. J. Benson} and \textit{J. F. Carlson} [Trans. Am. Math. Soc. 342, No. 2, 447-488 (1994; Zbl 0816.20044)]\ and \textit{D. J. Benson} and \textit{J. P. C. Greenless} [J. Pure Appl. Algebra 122, No. 1-2, 41-53 (1997; Zbl 0886.57022)]\ have studied homological duality properties for finite groups. Here the authors consider a virtual duality group \(\Gamma\) and show that the cohomology ring \(H^*(\Gamma)\) has strong ring theoretic duality properties, which combine the familiar homological duality of the (duality) subgroups of \(\Gamma\) of finite index with the duality enjoyed by the finite quotients. Both these two different types of duality may be obtained as special cases of the authors' result. The main theorem of the paper -- too complicated to be repeated here -- is expressed in terms of a spectral sequence involving the Grothendieck \(J\)-local cohomology where \(J\) is the ideal of positive degree elements in \(H^*(\Gamma)\). The authors give both a purely algebraic approach which applies to the situation over an arbitrary commutative Noetherian ring, and a topological approach for those virtual duality groups for which the duality is realized topologically (arithmetic groups, mapping class groups). The latter approach applies also to more general situations involving Lie groups and to generalized cohomology theories.
scientific article; zbMATH DE number 1038262
scientific article