The Jespers-Van Oystaeyen conjecture
There has been some interest recently in studying certain properties of orders, in particular maximal orders, by introducing several ''class groups'' of the order. We mention Fossum's K-theoretic class group, Fröhlich's Picard groups, Chamarie's normalizing class group and also the centralizing class group. The obvious question that presses forward is: what can one say about the order if its central class group equals the class group of the center. The author establishes that this is almost always equivalent to the fact that the order is a reflexive Azumaya algebra; the exceptional cases given arise from inseparability phenomena in certain residue algebras. The construction of generalized Rees rings initiated by the reviewer allows thus to reduce the study of maximal orders over a Krull domain to the study of reflexive Azumaya algebras (an easy class of rings!?) over certain Rees-type extensions of the center which are rather well-understood by using graded techniques. The reader should note that Proposition 2 allows a much shorter and easy proof whereas the proof of Proposition 3 contains a (repairable if one puts in some extra work) error at the spot where the author refers to the theory of primes of rings on p. 42. Of course, this can only enhance the author's conviction, expressed in the final line of his introduction, that development of this approach will lead to a better understanding of maximal orders.
scientific article; zbMATH DE number 3851303
scientific article