##
**Hopf algebras and Galois theory.**
*(English)*
Zbl 0197.01403

Lecture Notes in Mathematics. 97. Berlin-Heidelberg-New York: Springer-Verlag. ii, 133 p. (1969).

The Galois theory, as known for separable field extensions, certainly is one of the most appealing and powerful methods in algebra. An analogue of this exists in the case of inseparable field extensions, namely the Jacobson-Bourbaki theory. However, it turns out that the obvious generalization of the theory is false for arbitrary (inseparable) field extensions [L. Bégueri, Bull. Sci. Math., II. Ser. 93, 89–111 (1969; Zbl 0186.35502), p. 106]. However in various cases one likes to have methods available which mimick the fundamental Galois theorem. The present set of notes is fully devoted to the subtleties of such a question.

The volume contains three parts:

Chapter I: S. U. Chase – Galois objects;

Chapter II: S. U. Chase and M. E. Sweedler – Hopf algebras and Galois theory of rings;

Chapter III: S. U. Chase – Galois objects and extensions of Hopf algebras.

As it is not possible to condense the contents of 130 pages of rather technical results into a brief review, we shall only indicate briefly the trend of the arguments. In the first chapter we find the definition of a Galois object (it does not destroy properties of other objects by taking products, “faithful”, and it operates principally homogeneous); this notion extends the known one in case of separable field extensions. In the second chapter we find an application in commutative ring theory, where the difficulties alluded above apparently are overcome by studying only admissible Hopf subalgebras.

In the third chapter we find a classification of Galois objects under a finite commutative flat \(R\)-group scheme in terms of an extension group of sheaves in the Zariski topology on the category of schemes over \(\mathrm{Spec}\,R\). This result is not surprising, as it is known that in case of smooth linear kernels the usual \(\mathrm{Ext}^1\)-group coincides with the extension group of the Zariski sheaves (cf. proposition 17.4 in the reviewer’s [Commutative group schemes. Lecture Notes in Mathematics. 15. Berlin etc.: Springer-Verlag (1966; Zbl 0216.05603)]). In case the rank of the group scheme is prime to the characteristic one recaptures the classical Kummer theory. It could be interesting to see more connections with existing theories, in order to get a feeling for the strength of the methods exposed, and one should like to see new results obtained by the machinery described in these notes.

The volume contains three parts:

Chapter I: S. U. Chase – Galois objects;

Chapter II: S. U. Chase and M. E. Sweedler – Hopf algebras and Galois theory of rings;

Chapter III: S. U. Chase – Galois objects and extensions of Hopf algebras.

As it is not possible to condense the contents of 130 pages of rather technical results into a brief review, we shall only indicate briefly the trend of the arguments. In the first chapter we find the definition of a Galois object (it does not destroy properties of other objects by taking products, “faithful”, and it operates principally homogeneous); this notion extends the known one in case of separable field extensions. In the second chapter we find an application in commutative ring theory, where the difficulties alluded above apparently are overcome by studying only admissible Hopf subalgebras.

In the third chapter we find a classification of Galois objects under a finite commutative flat \(R\)-group scheme in terms of an extension group of sheaves in the Zariski topology on the category of schemes over \(\mathrm{Spec}\,R\). This result is not surprising, as it is known that in case of smooth linear kernels the usual \(\mathrm{Ext}^1\)-group coincides with the extension group of the Zariski sheaves (cf. proposition 17.4 in the reviewer’s [Commutative group schemes. Lecture Notes in Mathematics. 15. Berlin etc.: Springer-Verlag (1966; Zbl 0216.05603)]). In case the rank of the group scheme is prime to the characteristic one recaptures the classical Kummer theory. It could be interesting to see more connections with existing theories, in order to get a feeling for the strength of the methods exposed, and one should like to see new results obtained by the machinery described in these notes.

Reviewer: Frans Oort (Utrecht)