On Carlson's conjecture for cohomology rings of modules
Let K be an algebraically closed field of characteristic \(p>0\) and G be a finite group. For a finitely generated left KG module M denote the cohomology ring \({\mathcal E}_ G(M)=Ext^*_{KG}(M,M)\simeq H^*(G,End_ K(M))\). The purpose of this paper is to give a complete proof to a conjecture that \textit{J. F. Carlson} raised in two papers on the cohomology ring of modules [published in J. Pure Appl. Algebra 36, 105-121 (1985; Zbl 0565.20003) and 44, 85-97 (1987; Zbl 0617.20029)]. Carlson proved in the case when M was induced from a linear KH module for a normal subgroup H of G that every maximal two sided ideal contains the kernel of the restriction to some cyclic shifted subgroup. The principal method the author uses is to regard \({\mathcal E}_ G(M)\) as a finitely generated module over \({\mathcal E}_ G(K)\). As an application it is proved a theorem of Avrunin and Scott which asserts the equality of the module variety and the rank variety.
scientific article; zbMATH DE number 4110880