Class groups of maximal orders over Krull domains
Let R be a (commutative) Krull domain with field of fractions K and let \(\Lambda\) be a maximal order over R in some central simple algebra \(\Sigma\) over K. The normalizing classgroup of \(\Lambda\), \(C\ell(\Lambda)\), is studied by geometric tools. (\(C\ell(\Lambda)\) is expressed in terms of Picard groups of certain varieties.) To this end Weil and Cartier divisors are introduced in the case of ''noncommutative affine schemes'' associated with \(\Lambda\). Since all maximal orders are PI-rings this can be done by extending divisors of the central scheme. The methods are applied to obtain the following results: - a characterization of those locally factorial Krull domains R for which all maximal orders in \(M_ n(K)\) are conjugated, - the property that every torsion element of the Picard group of R is killed in the class group of a certain \(M_ n(R),\) - a proof that \(C\ell(\Lambda)\) is a direct summand of \(C\ell(\Lambda t).\) In the last section some results of the central classgroup of \(\Lambda\) are given. Here the approach is more ring theoretical.
scientific article; zbMATH DE number 3849419
scientific article