The Picard group of noncommutative rings, in particular of orders

The strucrure of the Picard group of not necessarily commutative rings, and specifically of orders, and its relation to the automorphism group are studied, mainly with arithmetic applications in mind. This paper is concerned with the Picard group, Pic (A), of a noncommutative (i.e. not necessarily commutative) ring A with identity, defined via the tensor product of invertible bimodules (see e.g. [Bl] or [B2]). Although some general results on Pic (A) are known, relatively little systematic work has been done so far. The real interest of the noncommutative theory has until now been the special case of Azumaya (central separable) algebras, when in fact Pic (A) coincides with Pic of the commutative ring cent (A) (the centre of A) (see [RZ]). On the other hand our principal interest lies in orders. But although our strongest results come in this case, some of these extend without too much weakening to a wider class of rings. In fact in the early part of the paper the treatment is completely general. Although most of the present paper is algebraic in spirit, the whole work was done with applications to the arithmetic of noncommutative orders over the ring Z of integers in mind. Our theory then becomes a tool both for certain local as well as for noncommutative "local-global" problems, and it leads up to the consideration of arithmetic subgroups in certain algebraic groups. Moreover the explicit computations for integral group rings given at the end of this paper are really number theoretic, i.e. depend crucially on Z being the base ring. For the general theory one can, without loss of generality, view the ring A as an algebra over some commutative ring R and look at Pic„ (A), the group given by bimodules with R acting the same on both sides. One can always take R = Z, noting that Pic (A) = Pic7(A). For theoretical reasons and for applications it is however the normal subgroup Pic , . .(A) = Picent(A) which is the ° r cent(A ) really interesting object, even when one considers R-orders, and R /= cent (A). Received by the editors July 19, 1972. AMS (MOS) subject classifications (1970). Primary 16A54, 16A18, 16A26, 16A72. (')Most of this work was done while the author was Visiting Professor in the University of Arizona at Tucson and was partly supported by the National Science Foundation. 1 Copyright © 1973, Z\merican Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use