On class groups of lattices
The genus of an order over a Dedekind domain in a central simple algebra can be divided in so-called spinor genera by a local conjugation condition on the completions of the order at the prime ideals of the centre. In this paper the author aims to derive general formulae for SG(\(\Lambda)\), the group of spinor genera, similar to the formulae for the locally free class groups, using a formalism introduced by A. Fröhlich, also M. Kneser, in the computation of class groups of algebras, resp. spinor class groups of lattices on quadratic spaces. In Section 3 of the paper the author extends the Fröhlich-Jacobinski formula for the locally free class group. In Section 4 the author obtains a Fröhlich-Jacobinski formula for the group of spinor genera in the genus of a lattice on a quadratic space. In the particular case where L is a maximal R-lattice on a regular quadratic space A the spinor genera group may be expressed in terms of the ray class group \(Cl_ q(R)\) of R relative to q, in fact \(SG(L)=Cl_ q(R)/G(A)Gl_ q(R)^ 2\) where G(A) is generated by the classes of prime ideals \(\rho\) of R dividing the discriminant of A; this corresponds to R. Swan's formula for class groups of maximal orders.
scientific article; zbMATH DE number 4000143
scientific article