Every projective Schur algebra is Brauer equivalent to a radical Abelian algebra. (English) Zbl 1147.16021
Let \(K\) be a field, \(A\) a (finite-dimensional) central simple \(K\)-algebra, \(K^*\) the multiplicative group of \(K\), \(L/K\) an arbitrary field extension, and \(\text{Rad}_K(L)\) the subgroup of those elements of \(L^*\) whose orders modulo \(K^*\) are finite. We say that \(L/K\) is radical, if \(L=K(\text{Rad}_K(L))\). The algebra \(A\) is called radical, if it is a crossed product \(A=(L/K,\alpha)\), where \(L/K\) is a radical Galois extension and \(\alpha\in\text{Rad}_K(L)\); when this holds and the Galois group \(G(L/K)\) is Abelian, \(A\) is said to be radical Abelian. We say that \(A\) is a projective Schur algebra, if it is spanned over \(K\) by a group of units that is finite modulo \(K\); this holds if and only if \(A\) is a homomorphic image of a twisted group algebra \(K^tG\), for some finite group \(G\) and \(2\)-cocycle \(t\).
The similarity classes of projective Schur \(K\)-algebras form a subgroup of the Brauer group \(\text{Br}(K)\), called the projective Schur subgroup of \(K\), and denoted by \(\text{PS}(K)\). It is known that \(\text{Br}(K)=\text{PS}(K)\), if \(K\) is a global or local field (Lorenz-Opolka, 1978), and also, if \(K\) contains a primitive \(n\)-th root of unity, for each \(n\in\mathbb{N}\) (Merkurjev-Suslin, 1982). Generally, however, we have \(\text{PS}(K)\neq\text{Br}(K)\), as shown by J. Sonn and the first author [J. Algebra 178, No. 2, 530-540 (1995; Zbl 0852.16011)], when \(K\) is a rational function field in one indeterminate over a finitely generated extension \(k\) of a global field \(k_0\).
The purpose of the paper under review is to complete the proof of the analogue to the Brauer-Witt conjecture for projective Schur algebras, which states that every projective Schur \(K\)-algebra is similar to a radical Abelian \(K\)-algebra. When \(\text{char}(K)\neq 0\), the conjecture has been proved in an earlier paper by J. Sonn and the first author [J. Algebra 239, No. 1, 356-364 (2001; Zbl 1012.16020)], so here they consider only the case of \(\text{char}(K)=0\). As a consequence, the paper obtains a characterization of \(\text{PS}(K)\) in terms of Galois cohomology. An important part of the proof is contained in Proposition 7, which uses character theory of projective representations.
The similarity classes of projective Schur \(K\)-algebras form a subgroup of the Brauer group \(\text{Br}(K)\), called the projective Schur subgroup of \(K\), and denoted by \(\text{PS}(K)\). It is known that \(\text{Br}(K)=\text{PS}(K)\), if \(K\) is a global or local field (Lorenz-Opolka, 1978), and also, if \(K\) contains a primitive \(n\)-th root of unity, for each \(n\in\mathbb{N}\) (Merkurjev-Suslin, 1982). Generally, however, we have \(\text{PS}(K)\neq\text{Br}(K)\), as shown by J. Sonn and the first author [J. Algebra 178, No. 2, 530-540 (1995; Zbl 0852.16011)], when \(K\) is a rational function field in one indeterminate over a finitely generated extension \(k\) of a global field \(k_0\).
The purpose of the paper under review is to complete the proof of the analogue to the Brauer-Witt conjecture for projective Schur algebras, which states that every projective Schur \(K\)-algebra is similar to a radical Abelian \(K\)-algebra. When \(\text{char}(K)\neq 0\), the conjecture has been proved in an earlier paper by J. Sonn and the first author [J. Algebra 239, No. 1, 356-364 (2001; Zbl 1012.16020)], so here they consider only the case of \(\text{char}(K)=0\). As a consequence, the paper obtains a characterization of \(\text{PS}(K)\) in terms of Galois cohomology. An important part of the proof is contained in Proposition 7, which uses character theory of projective representations.
Reviewer: Ivan D. Chipchakov (Sofia)
