On codimensions of maximal ideals in cohomology rings (Q1310508)

Let \(G\) be a finite group, \(K\) a field of characteristic \(p > 0\) and \(A\) a finitely generated left \(KG\)-module. The author [J. Pure Appl. Algebra 59, No. 3, 265-277 (1989; Zbl 0678.16019)] proved that if \(K\) is algebraically closed then every maximal ideal in \(E_ G(A) = \text{Ext}^*_{KG}(A,A)\) contains the kernel of the restriction map to some cyclic shifted subgroup. This was conjectured by \textit{J. F. Carlson} [ibid. 36, 105-121 (1985; Zbl 0565.20003) and 44, 85-97 (1987; Zbl 0617.20029)]. Here the author extends this result to: Theorem A. If \(K\) is algebraically closed then for a maximal ideal \(M\) in \(E_ G(A)\), there exists a cyclic shifted subgroup \(U\) and a maximal ideal \(N\) in \(E_ U(A)\) such that \(M\) contains \(\text{res}^{-1}_{G,U}(N)\). Moreover, using Theorem A he proves: Theorem B. If \(S\) is a simple \(E_ G(A)\)- module, then \(\dim_ K S \leq \dim_ K A\).