Our main result states that for a commutative ring R and a finite abelian group G the following conditions are equivalent: (a) Gal(R,G)=Gal (R[X],G), i.e. every commutative Galois extension of R[X]with Galois group G is extended from R. (b) The order of G is a non-zero-divisor in R/Nil(R). The proof uses lifting properties of Galois extensions over Hensel pairs and a “Milnor-type” patching theorem.

Abelsche Galoiserweiterungen von R[X]

Semigroup rings

We determine, for which x∈R* the Kummer extension K(Pn√x) is unramified over K, where K is a local field of char. 0 and res.char. p, ζpn∈K, and R is the valuation ring of K. This builds on techniques of Hasse. Actually, a much more general result concerning cyclic Galois extensions of commutative rings is proved.

Unramified Kummer extensions of prime power degree.

Algebraic K-theory

Preliminaries.- Central separable algebras and the brauer group.- Galois theory.- The six-term exact sequence.- Applications and remarks.

Separable algebras over commutative rings

Anneaux locaux henséliens

Introduction. The main objects of study in this paper are the commutative separable algebras over a commutative ring. Noncommutative separable algebras have been studied in [2]. Commutative separable algebras have been studied in [1] and in [2], [6] where the main ideas are based on the classical Galois theory of fields. This paper depends heavily on these three papers and the reader should consult them for relevant definitions and basic properties of separable algebras. We shall be concerned with commutative separable algebras in two situations. Let R be an arbitrary commutative ring with no idempotents except 0 and 1. We first consider separable P-algebras that are finitely generated and projective as an P-module. We later drop the assumption that the algebras are projective but place restrictions on P — e.g., P a local ring or a Noetherian integrally closed domain. In §1 we give a proof due to D. K. Harrison that any finitely generated, projective, separable P-algebra without proper idempotents can be imbedded in a Galois extension of P also without proper indempotents. We also give a number of preliminary results to be used in later sections. In §2 we generalize some of the results about polynomials over fields to the case of aground ring P with no proper idempotents. We show that certain polynomials (called \"separable\") admit \"splitting rings\" which are Galois extensions of the ground ring. We apply this to show that any finitely generated, projective, separable homomorphic image of R[X] has a kernel generated by a separable polynomial. In §3 we restrict our attention to separable algebras over a local ring. (The term \"local\" will not imply any finiteness conditions.) In certain cases every finitely generated, separable algebra is a homomorphic image of a finitely generated, projective, separable algebra. This gives an external characterization of the separable algebras. In §4 we consider the internal structure of seperable algebras over a Noetherian integrally closed domain. We show that a finitely generated separable algebra is the direct sum of projective, separable domains containing the ground ring and an algebra that is separable but not faithful over the ground ring. For the case of a Dedekind domain we can obtain specific infor-

/pdf/separable-algebras-over-commutative-rings-4xqd755orz.pdf

Abelsche Galoiserweiterungen von R[X]

Citations

Semigroup rings

Unramified Kummer extensions of prime power degree.

References

Algebraic K-theory

Separable algebras over commutative rings

Anneaux locaux henséliens

Separable algebras over commutative rings

Azumaya algebras with involution