The separable Galois theory of commutative rings (Q2841480)

This book provides a complete and self-contained account of the Galois theory of commutative rings from the viewpoint of categorical classification theorems and using solely the techniques of commutative algebra. The first edition of this book was published in 1974 [\textit{A. R. Magid}, The separable Galois theory of commutative rings. New York: Marcel Dekker, Inc. (1974; Zbl 0284.13004)].NEWLINENEWLINEContentsNEWLINENEWLINEIntroductionNEWLINENEWLINE1. SeparabilityNEWLINENEWLINE2. Idempotents and Profinite SpacesNEWLINENEWLINE3. The Boolean SpectrumNEWLINENEWLINE4. Galois Theory over a Connected BaseNEWLINENEWLINE5. Separable Closure and the Fundamental GroupoidNEWLINENEWLINE6. Categorical Galois Theory and the Galois CorrespondenceNEWLINENEWLINEIn the first chapter the author starts by looking at separable fields. Then he defines separable ring extensions and the author turns to a separable characterization in terms of square-zero extensions. Finally, the author considers the case of polynomial extensions as an extended example.NEWLINENEWLINEProfinite spaces, covering spaces and profinite group actions are investigated in the second chapter.NEWLINENEWLINEThe Boolean spectrum is investigated in the third chapter. The Boolean spectrum is about the space of connected components of the spectrum a commutative ring and the canonical sheaf on it. The topology on that space is determined by the Boolean algebra of idempotents of the ring. The main results obtained show basically that any date of finite type over the stalks can be lifted, and that objects agree if they agree at every stalk.NEWLINENEWLINEThe next chapter considers separable extensions when there are no nontrivial idempotents in the base. Let \(R\) be a connected commutative ring. Let \(S=C(X,S_0)\) where \(X\) is a profinite space and \(S_0\) is an infinite Galois extension of \(R\). Let \(G=\mathrm{Aut}(S_0)\) and let \(G(S/R)\) denote the groupoid \(X\otimes X\otimes G\). Then there is inverse bijection between the set of all closed subgroupoids of \(G(S/R)\) with are intersections of open-closed subgroupoids and the the set of all locally separable extensions of \(R\) in \(S\).NEWLINENEWLINEThe next chapter constructs the categorical equivalence for an arbitrary commutative rings. The author constructs for a commutative ring \(R\) with separable closure \(S\) a contravariant natural equivalence between the category \(\mathcal{S}_{\mathrm{cls}}(R)\) of componentially locally strongly separable extension of \(R\) and the category \(\mathcal{M}=\mathcal{M}_{\mathrm{prf}}(X(S\otimes_R))\) of (pro-relatively finite) profinite sets, \(\Pi_1=\Pi_1(R,S)\Pi=X(S\otimes_R S)\) carries its standard groupoid structure. This categorical equivalence does not automatically produce a correspondence between sub ring extensions of an extensions. For that the author have to look at relative closures, which is what is considered in Chapter~6. The main results is the following:NEWLINENEWLINE Theorem. Let \(R\) be a commutative ring and let \(S\) be an autosplit componentially locally strongly separable extension of \(R\). Then there is a one-one correspondence between the set of componentially locally strongly separable of \(R\) contained in \(S\) and the set of quotients of \(\mathcal{G}(S/R)\) in \(\mathcal{M}_{prf}(\mathcal{G}(S/R))\).