The cohomology ring of a module
Let G be a finite group, K an algebraically closed field of characteristic \(p>0\) and M a finitely generated KG-module. It was shown by the author [in Lect. Notes Math. 903, 62-67 (1981; Zbl 0475.20040)] that an element of \(Ext^{\lambda}_{KG}(M,M)\) is nilpotent iff its restriction to every elementary abelian p-subgroup of G is nilpotent. Now let \(G=<x_ 1,x_ 2,...,x_ n>\) be an elementary abelian p-group, \(\alpha =(K_ 1^{\alpha},K_ 2^{\alpha},...,K_ n^{\alpha})\in K^ n\) with \(\alpha\) \(\neq 0\) and \(u_{\alpha}=1+\sum_{i}K_ i^{\alpha}(x_ i-1)\). Then \(u_{\alpha}\) is a unit of KG of order p. If \(\{\alpha_ 1,\alpha_ 2,...,\alpha_ m\}\) is a set of linearly independent elements of \(K^ n\), then the group \(H=<u_{\alpha_ 1},...,u_{\alpha_ m}>\) is called a shifted subgroup of KG. The embedding KH\(\to KG\) induces a ''restriction'' map on cohomology. Here the author shows that an element of \(Ext^{\alpha}_{KG}(M,M)\) is nilpotent iff its restriction to certain \(<u_{\alpha}>'s\) is nilpotent. He then characterizes the radical of \(Ext^*_{KG}(M,M)\) in terms of restrictions to shifted subgroups of KG. These results are then used to give a new proof of a theorem of \textit{G. S. Avrunin} and \textit{L. L. Scott} [Invent. Math. 66, 277-286 (1982; Zbl 0489.20042)] which states that the cohomology variety W(M) of M is equal to the rank variety V(M) of M. (W(M) is the zero set of an ideal in a polynomial subring of \(Ext^*_{KG}(K,K)\) and \(V(M)=\{0\}\cup \{\alpha \in K^ n|\) \(M_{<u_{\alpha}>}\) is not a free \(K<u_{\alpha}>\)-module\(\}\).) The main theorem was proved by the author for \(p=2\) [in J. Algebra 85, 104- 143 (1983; Zbl 0526.20040)]. Here the proof is different and uses a nontrivial spectral sequence argument. The author adopts the Avrunin- Scott point of view in regarding KG as the restricted enveloping algebra of a commutative Lie algebra. The corresponding coalgebra structure then gives a cup product action of \(Ext^*_{KG}(K,K)\) on \(Ext^*_{KG}(M,M)\) that commutes with restrictions to shifted subgroups. Note that the product in \(Ext^*_{KG}(M,M)\) is independent of the coalgebra structure since it coincides with the Yoneda product. An example of an indecomposable module M such that \(Ext^*_{KG}(M,M)/Rad Ext^*_{KG}(M,M)\) is not commutative is given. The author has shown [in Proc. Lond. Math. Soc., III. Ser. 47, 480-492 (1983; Zbl 0555.20030)] that if \(G=SL(2,p^ n)\) and M an irreducible KG-module, then the ring \(Ext^*_{KG}(M,M)/Rad Ext^*_{KG}(M,M)\) is commutative.
restricted enveloping algebra of a commutative Lie algebra
scientific article; zbMATH DE number 3901008
scientific article