Linear tori with the action of finite groups (Q1909779)

The main object of the paper is a linear representation \(V\) of a finite group \(G\) with an additional structure, namely with a \(G\)-stable torus \(T\) embedded into \(V\) as an open subset which is called a linear torus. If the action of \(G\) on \(T\) is free, \(T\) is called a linear \(f\)-torus. One can introduce a natural partial order on the set of linear tori: \(T \geq T'\) if there is a \(G\)-equivariant map \(T \to T'\). It turns out that in the set of linear \(f\)-tori there is only one maximal equivalence class with respect to this relation, and it can be distinguished by the following condition: the image of \(K_0 (T/G)\) in the cohomology of \(T/G\) is trivial. The author uses this result to define partially stabilized cohomology \(H^*_{\text{par}} (G,F)\) with coefficients in a discrete module \(F\) as the image of \(H^* (G,F)\) in \(H^*(T/G, F)\) where \(T\) is a linear \(f\)-torus from the maximal equivalence class. This notion was introduced earlier [the author, Mat. Sb. 183, No. 5, 3-28 (1992; Zbl 0789.14022)] as an attempt to give a more computable analog of stabilized cohomology.