Crossed Product Orders and Non-commutative Arithmetic

Abstract

Let Λ = (S/R, h) be a crossed product order (CPO) in the crossed product algebra A = (L/K, h) with integer factor set h. Studying the chain of orders Λ0 ≔ Λ, Λi + 1 ≔ Ol (rad Λi) we give, in the local case, a measure of the deviation from the CPO to its hereditary hull and can thus classify all hereditary local CPOs. In the local case there exists a unique optimal CPO and we show how to optimize a given factor set. If the extension L/K is also tamely ramified we can compute the Schur index of A by the values of the factor set. Studying the semilocal and the global case we give necessary criteria for hereditary CPOs and solve the heredity question completely for cyclic CPOs.