Regional Conference Series in Mathematics. 94. Providence, RI: American Mathematical Society (AMS). vii, 119 p. $ 22.00 (1999).
The author’s aim in these lectures is to present the foundations of the modern theory of division algebras and provide a solid background for the massive research done in recent years. He begins with definitions and examples but soon introduces Azumaya algebras and many of the later results are stated in this context. Thus he describes the `form of matrices’ obtained by making a faithfully flat extension of the ground ring. The Brauer group is defined for any commutative ring $R$, though for its more recondite properties $R$ is taken to be a field. Next crossed products are introduced and are used to describe the equivalence between Brauer groups and certain Galois cohomology groups. This leads to the corestriction and the Merkurjev-Suslin theorem which is quoted without proof. After a brief discussion of orders, lattice methods and ramification the author comes to lifting properties and specializations, leading to the notion of a generic division algebra, which is discussed in some detail and is related to the Brauer-Severi variety. The style of the whole is very informal, not to say casual. This sometimes makes for better understanding but occasionally leads to confusion (much like a real lecture). Only a basic background is expected from the reader, e.g. some proofs depending on étale cohomology are omitted, with just a reference. A serious lack is the omission of an index and a list of notations, but this the reader can make for himself. In all this is a useful survey of the present state of the subject, going well beyond most other texts in this field.