Cohomology of Hopf algebras and the Clifford's extension problem (Q1920004)

The purpose of this paper is to generalize some results on Hopf-Galois extensions for algebraically closed fields [\textit{G. Militaru} and \textit{D. Ştefan}, Commun. Algebra 22, No. 14, 5657-5678 (1994; Zbl 0816.16029)] to any fields. Let \(H\) be a Hopf algebra over a field \(k\), \(A/B\) a faithfully flat \(H\)-Galois extension and \((M,\mu)\) a right \(B\)-module which is supposed to be finite-dimensional. In Section 1, the author proves that the right ideal generated by the annihilator \(I\) of \(M\) is a two-sided ideal of \(A\) and a right coideal, so \(A/IA\) is a comodule algebra and its coinvariant subalgebra is isomorphic with \(B/I\), and the extension \(A/AI\subseteq B/I\) is faithfully flat \(H\)-Galois. Moreover, there exists an \(A\)-extension for \(M\) iff there exists an \(A/AI\)-extension for the \(B/I\)-module \(M\). In Section 2, the author proves that an \(H\)-Galois extension \(A/B\) with \(B\) a simple ring is always an \(H\)-crossed product. Then it is shown that \(A\) and \(E\otimes M_n(k)\) are isomorphic as comodule algebras where \(n=\dim({_DM})\), \(E=\text{End}_A(M\otimes_BA)\) and \(D=\text{End}_B(M_B)\). As a corollary, the Galois structures of \(A\) and \(E\) give the same Ulbrich-Miyashita action on the center \(Z\) of \(B\). For a simple subalgebra \(R\) of \(B\), the centralizer of \(R\) in \(A\) is an \(H\)-Galois extension of the centralizer of \(R\) in \(B\). In Section 3, the Clifford obstruction is defined if there exists a central division algebra \(D_0\) over \(F\) such that \(D\cong Z\otimes_FD_0\) where \(F\) is the fixed field of the Ulbrich-Miyashita action on \(Z\). For such a division algebra, \(\Omega_M/Z\) is defined to be the centralizer of \(D_0\) in \(E\). As \(Z\) is commutative, \(\Omega_M\) is associated to the Clifford obstruction \(\omega_M\in H^2(H,Z)\). In the main results, the connection between \(\omega_M\) and the extension problem is studied. Finally, it is shown that for a large enough class of pointed Hopf algebras, the Clifford obstruction is defined. As a final remark, in the particular case \(H=k[G]\) for \(G\) a finite group, the author obtains Hopf algebraic proofs of some well known results concerning strongly graded rings and the extension problem for these rings.