Azumaya algebras which are not smash products
Let R be a commutative ring, H and R-Hopf algebra, finitely-generated and projective as R-module, \(H^*\) the dual Hopf algebra. Let Gal(H) be the set of isomorphism classes (as R-algebras and H-modules) of Galois H- extensions. There is a smash-product map {\#}: Gal(H)\(\times G(H^*)\to Az(R)\) to the set of isomorphism classes of Azumaya R-algebras. Over number fields, every Azumaya algebra is isomorphic to such a smash product. This paper indicates the failure of this result over number rings. Thus, let R be the ring of integers in a number field K. For each R-Hopf algebra of rank 2, {\#} maps into \(Az_ 2(R)\), the set of isomorphism classes of rank 4 Azumaya R-algebras. The author's main theorem is as follows: Let n be an even integer. With at most finitely- many exceptions, for every totally real number field K of dimension n over the rationals, there exists an element of \(Az_ 2(R)\) which is not representable as a smash-product. This theorem applies to real quadratic fields. For \(K=Q(\sqrt{p})\), p a prime, he obtains the number of isomorphism classes of Azumaya R-algebras of rank 4 which are smash- products (in all cases, it is \(\leq 3)\), and he shows that there are rank 4-Azumaya algebras which are not smash-products for all \(p>3\).
scientific article; zbMATH DE number 4156677
scientific article