On trivialization of discriminant algebras of hyperbolic quadratic modules (Q1943368)

The author considers discriminant algebras of hyperbolic quadratic modules. He pursues their trivialization. He gives a critical overview of the topic as it developed. The starting point is N.~Bourbaki (1959, 1970), followed by contributions from M.-A.~Knus (1991), U.~Haag (1991), more importantly by O.~Loos (1997) and recently by H.~Ikai (2010, 2011). Clifford algebras, Pfaffians, and the discriminant are some of the important topics that are involved in the theory. The two main results that the author proves are the following two theorems: 1. Let \(M\) be a finitely generated projective module and let \(\Phi\) be a bilinear form on it. For elements \(v,w\) in \(\Lambda^2(M^\ast)\), each \(\kappa_{\Phi+\tilde{v},\Phi+\tilde{w}}^{(n)}\) is equal to the degree-\(2n\)-component of the image of \(\exp(a_\Phi)\wedge(\kappa(v)-\kappa(w))\) under the isomorphism \(\Theta_\wedge\). 2. As a bilinear representative of the hyperbolic quadratic form \(x \oplus f \mapsto \langle x,f \rangle\) on \(P \oplus P^\ast\), choose \(\Phi(x \oplus f, y \oplus g) := \langle x,g \rangle\). For the bilinear form \(\Phi\) and for any \(v\) in \(\Lambda^2(M^\ast)\), one has \(z_{\Phi+\tilde{v}}(\omega_0) = \eta(v){\text{ Id}_{\Lambda+P} \oplus (1+\eta(v))\text{ Id}_{\Lambda-P}}.\)