On deformations of crossed products.
The subject of the paper under review is the crossed product algebra \(A*\Gamma\), where \(A\) is semisimple, finitely generated over its center, and \(\Gamma\) is a finite group. The authors give a necessary and sufficient condition in terms of the outer action of \(\Gamma\) on \(A\) for the existence of a multi-parametric semisimple deformation on the form \(A(\!(t_1,\dots,t_n)\!)*\Gamma\) (with the induced outer action). Furthermore, suppose that the base ring \(A\) is semisimple Artinian. Recall that a crossed product \(A*\Gamma\) induces an action of \(\Gamma\) on the set of simple components \(\Lambda=\{A_i\}_{i\in I}\) of \(A\). Let \(\{A_{i_j}\}_{j\in J}\) be a set of representatives for the orbits in \(\Lambda\) and let \(\Gamma_{i_j}\) be the stabilizer of \(A_{i_j}\) in \(\Gamma\). Then \(A*\Gamma\cong\prod_{j\in J}M_{n_j}(A_{i_j}*\Gamma_{i_j})\) with \(n_j=[\Gamma:\Gamma_{i_j}]\). Let \(H_{i_j}=\ker(\alpha_{i_j}\colon\Gamma_{i_j}\to\text{Out}(A_{i_j}))\) and let \(P_{i_j}\) be \(p_i\)-Sylow subgroups of \(H_{i_j}\) (in case \(\text{char}(Z(A_{i_j}))=p_i>0\)). The following conditions play an important role in the further investigation. (A) \(P_{i_j}\) is Abelian with normal complement in \(H_{i_j}\); (B) \(\text{rank}(P_{i_j})\leq p_i\)-degree of \(Z(A_{i_j})\). The following results are obtained. Theorem 1.6. Let \(R=A*\Gamma\) be a crossed product, where \(\Gamma\) is finite and \(A\) is semisimple finitely generated over its center. Assume (with the above notation) that each \(P_{i_j}\) is Abelian and has a normal complement in \(H_{i_j}\). Then there is a semisimple \(n\)-parameter homogeneous deformation \(R'=A(\!(t_1,\dots,t_n))*\Gamma\) for some integer \(n\geq 0\). Theorem 1.7. Let \(p(i)\) be the \(p_i\)-degree of \(Z(A_{i_j})\), then the minimal number of parameters is given by \(n_0=\max_{i\in I}\{1,\text{rank}(P_{i_j})-p(i)\}\). The authors also consider the so called twisting problem, which can be formulated as follows: Let \(A_\alpha^{f_0}*\Gamma\) be a given crossed product and assume the necessary conditions (for semisimplicity) (A) and (B) are satisfied. Does there exist a semisimple crossed product \(A_\alpha^{f'}*\Gamma\)? The twisting problem was solved by E. Aljadeff and D. J. S. Robinson in the case when \(A=K\) is a field. E. Aljadeff, Y. Ginosar and Á. del Río gave a solution in the case when \(\Gamma\) is cyclic and \(A\) is finitely generated as a module over its center. In this paper the authors show that the twisting problem has a positive answer in the case when \(A\) is finitely generated as a module over its center. Theorem 3.1. Let \(A_\alpha^{f_0}*\Gamma\) be a crossed product, where \(A\) is a semisimple algebra which is finitely generated as a module over its center. Assume the necessary conditions (for semisimplicity) (A) and (B) are satisfied. Then there exists a semisimple crossed product \(A_\alpha^{f'}*\Gamma\). Theorem 3.2. Let \(\Gamma\) be a finite group, \(K\) a field of characteristic \(p>0\), and \(\eta\colon\Gamma\to\Aut(K)\) an action of \(\Gamma\) on \(K\) with kernel \(H\). Let \([f]\in H^2(H,K^*)\) be any class. Then there is a class \([g]\in\text{Im}(\text{res}_H^\Gamma \colon H^2(\Gamma,K^*)\to H^2(H,K^*) )\) such that \(K^{f\cdot g}H\) is semisimple if and only if any \(p\)-Sylow subgroup \(P\) of \(H\) is Abelian with a normal complement in \(H\) and further the rank of \(P\) does not exceed the \(p\)-degree of \(K\) over \(K^p\).
scientific article; zbMATH DE number 5146861