Separable Algebras

The natural generalization of a separable finite extension of fields is the notion of a commutative separable algebra, that is, a commutative algebra \(A\) of finite type over a base field \(k\) that is isomorphic to a direct product of a finite number of finite separable field extensions of the field \(k\). This is equivalent to \(A\) being reduced, that is, \(A\) has no nilpotent elements. Commutative separable algebras play an important role in algebraic number theory and algebraic geometry, which somehow shaped the first definition of separability or its various equivalent formulations. For example, when \(A\) is a commutative algebra of finite type over a field k, then \(A\) is separable if and only if \(A\) is a projective module over the ring \(A\otimes_k A\), via the multiplication map \(A\otimes_k A\rightarrow A\). This module-theoretic characterization of commutative separable algebras is equivalent to the requirement that the induced diagonal morphism of affine schemes \(\text{Spec} (A)\rightarrow \text{Spec} (A\otimes_k A)\) is an open immersion.

From the many characterizations of commutative separable algebras, the module-theoretic is the one amenable to generalization to non-commutative algebras over a commutative ring \(R\): An \(R\)-algebra \(A\) is separable over \(R\) if the map \(A\otimes_k A^{o}\rightarrow A\) makes \(A\) a left projective \(A\otimes_k A^{o}\)-module, where \(A^{o}\) is the opposite ring of \(A\).

The book under review is a thorough introduction to separable algebras over a commutative ring. For completeness the book includes background material on rings and modules over commutative rings, and homological algebra, paying special attention to Morita equivalence, faithfully flat modules and faithfully flat descent, a non-commutative version of Nakayama’s lemma, the Wedderburn-Artin theorem, and Hochschild and Amitsur cohomologies, including the low degree cases when the coefficient groups are non-commutative. There is also a chapter on the divisor class group that includes a discussion of its functorial properties, for the cases when the class group is indeed a functor, for example when the morphism between the corresponding rings is faithfully flat and finite. The exposition on the review material is comprehensive, sometimes with complete proofs or with a precise reference in the bibliography, and comprises chapters 1, 2, 3, 5, and 6.

The theory of separable algebras occupies the remaining chapters. It starts in chapter four, with the definition of separable algebra over a commutative ring, recalled before, and its basic properties (behavior under base change and the transitivity of separability) and some characterizations of separable algebras, for example in terms of the centralizer functor. This chapter also treats the case when the ground ring is a field, characterizing central separable algebras over a field as central simple algebras. One important result is the unique decomposition of a separable algebra over a field as a finite direct product of matrix algebras over division algebras whose centers are separable field extensions of the base field. A direct consequence is the characterization of commutative separable algebras over a field mentioned at the beginning of this review.

Over an arbitrary commutative ring \(R\), central separable algebras are called Azumaya algebras, and these algebras are introduced in chapter 7, where we also find the definition of the Brauer group of isomorphism classes of Azumaya \(R\)-algebras in general, including some of its properties when the ground ring is a field. Further invariants attached to Azumaya algebras and properties of the Brauer group are included in chapter 11.

The derivative characterization of separability over a field is duly generalized in chapter 8, which introduces Kähler differentials and uses them to give criteria for separability of commutative algebras over a ring, including the Jacobian criterion. Derivations are also used to define and study differential crossed product algebras, which are immediately applied to construct Azumaya algebras in characteristic \(p\).

Adding a flatness property to a commutative separable algebra over a ring \(R\) introduces the étale \(R\)-algebras, and chapter 9 is devoted to obtain the main properties of these algebras. Since being flat is a local property, naturally some section of this chapter is devoted to some local criteria for flatness, and somehow starts to focus on local Noetherian rings, and in particular on Henselian local rings. For example, in chapter 10 it is proved that for a Henselian local ring \(A\) with residue field \(k\), the category of finite étale \(A\)-algebras is equivalent to the category of commutative separable \(k\)-algebras. Henselian local rings, henselizations and strict henselizations of local rings and their relations to Azumaya algebras make the bulk of chapter 10, which is supplemented by a section on étale cohomology in the down-to-earth Čech version, which is directly applied to the calculation of the Brauer group of a commutative Noetherian ring

Most concepts and generalizations introduced in these two chapters, and the last section of chapter four, originated in Grothendieck’s notions of étale coverings and the étale site introduced in SGA 1, which evolved to the notions of formally smooth, formally unramified, and formally étale in EGA IV. The treatment in these chapters follows Raynaud’s Anneaux locaux henséliens (Springer, LNM 169, 1970), but equally could have followed H. Kurke and G. Pfister’s Henselsche Ringe und Algebraische Geometrie (VEB Deutscher Verlag der Wissenschaften, 1975), but this reference is not included in the bibliography.

Chapter 12 is devoted to the Galois theory of extensions of commutative rings. The topics covered include Galois descent, the fundamental theorem of Galois theory, embeddings of separable algebras, separable closures and infinite Galois theory. One section is devoted to cyclic extensions of rings, with particular focus on Kummer and Artin-Schreier extensions.

Chapter 13, on Galois cohomology ties several of the topics covered on the previous chapter and the chapters on Azumaya algebras, since for a given commutative ring \(S\) and a finite group of \(G\) of automorphisms of \(S\), for a subring \(R\) of \(S^G\) the extension \(S/R\) is Galois if the trivial crossed product defined by these data is an Azumaya algebra over \(R\). The last chapter treats some additional topics such as Milnor’s version of the Mayer-Vietoris sequence associated to certain Cartesian square of commutative rings, with applications to computing some Brauer groups of rings, for example for the ring of integers of a global field.

The thorough and comprehensive treatment of separable, Azumaya, and étale algebras, Hensel rings, the Galois theory of rings, and Galois cohomology of rings makes the book under review an indispensable reference for the graduate student interested on these topics. As an added bonus, the book comes with a rich, 155 item, bibliography, well-chosen examples, calculations, and sets of exercises in each chapter, which make this book an excellent textbook for self-study or for a topics course on separable algebras.

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Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.