Cohomological invariants of algebras with involution
Let \(A\) be a finite-dimensional central simple algebra over a field of characteristic different from \(2\). An involution on \(A\) is a ring anti-automorphism of period \(2\). To every involution \(\sigma\) on \(A\), the author attaches the quadratic form \(T_\sigma\colon A\to F\) which maps \(x\in A\) to the reduced trace of \(\sigma(x)x\), where \(F\) is the subfield of the center fixed under \(\sigma\), and the subform \(T_\sigma^+\) which is the restriction of \(T_\sigma\) to the \(F\)-vector space of symmetric elements. The main purpose of the paper is to compute the determinant and Hasse invariant of these quadratic forms, which may be regarded as invariants of the involution \(\sigma\). If the involution is of the second kind, i.e., if the center \(Z\) of \(A\) is a quadratic extension of \(F\), the author also defines a new cohomological invariant \({\mathcal D}(A,\sigma)\) which she calls the determinant class of \(A\). This invariant lies in the quotient of the cohomology group \(H^2(F,Z^1_n)\) by an action of \(\mu_2\), the group of order \(2\), where \(n\) is the degree of \(A\) and \(Z^1_n\) is the \(F\)-algebraic group of \(n\)-th roots of unity of norm \(1\) in \(Z\). The points of \(Z^1_n\) over a separable closure of \(F\) are the \(n\)-th roots of unity, with a twisted Galois action. If \(n\) is even, there is a canonical map \(H^2(F,Z^1_n)\to H^2(F,\mu_2)\) which is compatible with the action of \(\mu_2\). The image of \({\mathcal D}(A,\sigma)\) under this map is called the determinant class mod \(2\) and denoted \(D(A,\sigma)\). This element is explicitly computed in some special cases, and it is related to the Hasse invariant of \(T_\sigma^+\). The determinants of \(T_\sigma\) and \(T_\sigma^+\) are shown to depend only on \(Z\), and the Hasse invariant of \(T_\sigma\) is trivial. If \(\sigma\) is of the first kind, i.e., if \(F\) is the center of \(A\), the author shows that the determinant of \(T_\sigma\) is trivial and that the determinant of \(T_\sigma^+\) is directly related to the determinant of \(\sigma\), as defined by Jacobson, see also \textit{M.-A. Knus, R. Parimala, R. Sridharan} [Bull. Soc. Math. Belg., Sér. A 43, No. 1/2, 89-98 (1991; Zbl 0757.12002)]. The Hasse invariants of \(T_\sigma\) and \(T_\sigma^+\) are given by explicit formulas which depend only on the degree of \(A\), the determinant of \(\sigma\) and the Brauer class of \(A\). The Hasse invariant of \(T_\sigma\) has also been computed by \textit{D. Lewis} [Exp. Math. 15, No. 3, 265-272 (1997; Zbl 0889.11011)], by a different method. In the final section, the author shows that the determinant class mod \(2\) of a central simple algebra of degree \(4\) with involution of the second kind is trivial if the algebra decomposes into a tensor product of two quaternion algebras stable under the involution. The converse has recently been proved in a joint work by the author and N. Karpenko.
scientific article; zbMATH DE number 1053885
scientific article