The center of the generic division ring and twisted multiplicative group actions
Let \(F\) be a field, \(L\) and \(K\) extensions of \(F\), and \(G\) a finite subgroup of their groups of \(F\)-automorphisms. We say that \(L\) and \(K\) are stably isomorphic (over \(F\)), if there exist \(G\)-trivial indeterminates \(x_1,\dots,x_r\), \(y_1,\dots,y_s\), such that \(L(x_1,\dots,x_r)\cong_FK(y_1,\dots,y_s)\) as \(F\)-algebras, and the isomorphism agrees with their \(G\)-actions; when this occurs, \(L\) is called stably rational over \(K\), if \(K\) is a subfield of \(L\). The paper under review studies the question of whether the centre \(C_n\) of the division ring of \(n\times n\) generic matrices is stably rational over \(F\), with particular attention to the special case where \(n\) is a prime number. Denote by \(F(M)\) the quotient field of the group algebra \(F[M]\), for every Abelian torsion-free group \(M\). It is known [see \textit{E. Formanek}, Linear Multilinear Algebra 7, 203-212 (1979; Zbl 0419.16010)] that for every \(n\in\mathbb{N}\), there is a \(\mathbb{Z}[S_n]\)-lattice \(G_n\), for which the fixed field \(F(G_n)^{S_n}\) under the action of \(S_n\) on \(F(G_n)\) is stably isomorphic to \(C_n\). The present paper shows that if \(p\) is prime, \(H\) is a subgroup of \(S_p\) of order \(p\), \(\Phi\) is the normalizer of \(H\) in \(S_p\), \(L\) is a finite \(S_p\)-module with a \(p\)-primary component isomorphic to \(\mathbb{Z}[S_p]\otimes_{\mathbb{Z}[\Phi]}(\mathbb{Z}_1/p^r\mathbb{Z}_1)\), and \(E\) is any extension of \(L\) by \(\mathbb{Z}[S_p]\), such that \(E\) is a \(\mathbb{Z}[S_p]\)-lattice, then the centre of the division ring of \(p\times p\) generic matrices over \(F\) is stably isomorphic to \(F(E)^{S_p}\). Also, the author proves that if \(A\) is the root lattice and \(L=F(\mathbb{Z}[S_p]/H)\), then there exists an action of \(S_p\) on \(L(\mathbb{Z}[S_p]\otimes_{\mathbb{Z}[H]}A)\) twisted by an extension \(\alpha\in\text{Ext}_{S_p}^1(\mathbb{Z}[S_p]\otimes_{\mathbb{Z}[H]}A,L^*)\) (corresponding to an element of the relative Brauer group \(\text{Br}(L/L^H)\)), such that \(L_\alpha(\mathbb{Z}[S_p]\otimes_{\mathbb{Z}[H]}A)^{S_p}\) is stably isomorphic to \(C_p\). The latter result represents a reduction on the problem.
scientific article; zbMATH DE number 1891325