The Separable Galois theory of commutative rings

"This book provides a thorough, self-contained explanation of Galois theory of commutative rings. Requiring some background in commutative algebra and algebraic geometry, the book gives complete proofs of the main results of separable Galois theory. This edition includes a new chapter that offers background on separability and a new chapter on categorical Galois theory that incorporates many developments in the field. The book also covers Boolean spectrum theory and the fundamental groupoid"-- "The first edition of this book appeared with a copyright date of 1974; arithmeti- cally inclined readers will note that this is exactly 40 years prior to the copyright date of this edition. It was my hope, with the earlier edition, to give an account of a subject which I termed \more or less complete". Of course no mathematics is ever complete. The subject itself moved on, most notably, in the work of George Janelidze on Galois theory in categories, a de nitive account of which may found in the book Galois Theories by Janelidze and Francis Borceux. An explanation of that theory will not be attempted here. Rather, I have the much more modest aim of revisiting that earlier work with whatever insights an additional four decades of doing mathematics can bring to bear. That means that virtually every result and every piece of exposition has been rewritten and recast, often to include additional generality, although, at least to the author, the logical arc of the earlier volume remains mostly intact. One exception to the previous assertion is that this volume include a self contained exposition of the theory of separable algebras. The excellent lecture notes Separable Algebras over Commutative Rings by DeMeyer and Ingraham which was used as a citation source for that theory in the rst edition remains in print. However, the likelihood that mathematicians today have seen that ma- terial, either in the cited work or another source, is small, warranting including an exposition in this volume, and given that, the temptation to explain my own understanding of separability was too great to resist. This book is, at heart, commutative algebra, the author, at heart, being a commutative algebraist"--