Posted by Jason Polak on 29. December 2016 · Write a comment · Categories: commutative-algebra · Tags:

Let $R$ be a commutative ring. We say that an $R$-algebra $A$ is separable if it is projective as an $A\otimes_R A^{\rm op}$-module. Examples include full matrix rings over $R$, finite separable field extensions, and $\Z[\tfrac 12,i]$ as a $\Z[\tfrac 12]$-algebra.

The 1970 classic Separable Algebras by deMeyer and Ingraham acquaints the reader with this important class of algebras from two viewpoints: the noncommutative one through structure theory and the Brauer group, and the commutative one through Galois theory.

This book accomplished the rare feat of keeping me interested; throughout its pages I found I could apply its results to familiar situations: Why are the only automorphisms of full matrix rings over fields inner? Why are such rings simple? What makes Galois theory tick? Separable Algebras explains with clarity how familiar algebra works through the lens of separable algebras.

The book starts with a prerequisites chapter on projective modules, Morita theory, and the Picard group. I already felt comfortable with most of this material, except Morita theory, the theorems of which I find rather easy to forget despite learning it on multiple occasions. This first chapter is handy as a reference for such forgotten facts.

Separable algebras are introduced in Chapter 2, along with their basic properties, central separable algebras (aka Azumaya algebras), and the Brauer group. This chapter has left me with the desire to find separable algebras everywhere since they have such nice properties. For example, the ideals of a central separable algebra are in bijective correspondence with the ideals of the center, which shows why full matrix rings over fields must be simple. This chapter is filled with results I feel I could actually use in my own research and explorations.

The third chapter covers the Galois theory of commutative rings with no nontrivial idempotents. This is an analogue to the theory of fields: there is a bijective correspondence between subgroups of a Galois group and sub-separable algebras of a given Galois extension of a commutative ring. This chapter not only provides very useful tools for the practicing algebraist, it also is very satisfying in that it increased my understanding of the Galois theory of fields. While separable algebras is certainly not the first route one would take to the Galois theory of fields, it is one of the most natural.

Chapter 4, consisting of fifteen pages, is one I did not examine very closely. It is the construction of an exact sequence relating various group cohomology groups to projective class groups and relative Brauer groups. Although an interesting and useful sequence, I feel comfortable taking it on faith, with the intention of looking at the gritty details in case I really need them.

The final Chapter 5 of the book is a short exposition of some structure theory and the computation of the Brauer group of a Dedekind domain, along with some at the time unsolved questions in the field.

I could find very few flaws with Separable Algebras: there are a few typographical errors, but none of them major. The authors seem to have an obsession with the dual basis formulation of projectivity where sometimes other equivalent formulations would lead to slicker proofs, though the reader should find it instructive to find such proofs. In Chapter 3, they use the word “finite” for “finitely generated as a module” without actually defining it. Such usage may be familiar to algebraic geometers but to me, it is one that is continually irksome.

And what of the other books or notes that cover similar material? There is Andy Magid’s nice book The Separable Galois Theory of Commutative Rings, a second edition of which just came out in 2014. Magid’s book focuses on the the Galois theory of commutative rings. Unlike the title under review, Magid goes further and does Galois theory with arbitrarily many idempotents and thus requires the use of profinite spaces, the Boolean spectrum, and scary groupoids. Magid’s book is fairly self-contained, but is more advanced and requires more generality that is not necessary if one doesn’t need to consider idempotents.

Then there’s also H.W. Lenstra’s notes Galois Theory for Schemes, available for free on the internet. Lenstra covers Galois theory not only for commutative rings, but also for schemes. His book contains only the bare minimum about separable algebras (mostly as exercises) in order to present the definition and basic properties of the etale fundamental group of a scheme. As Lenstra’s notes are quite terse, I would strongly suggest first looking at Separable Algebras first, if one is interested in schemes.

In any case, the best bet for a newcomer to separable algebras is deMeyer and Ingraham’s book, which should prepare the reader for anything more advanced. Highly recommended.

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