Stephen U. Chase

A theorem of Harrison, Kummer theory, and Galois algebras

Let R be a field and S a separable algebraic closure of R with galois group R . In [8] Harrison succeeded in describing R / ′ R in terms of R only. More precisely, he constructed a certain complex ( R, Q/Z ) and proved Hom c , where Hom c denotes continuous homomorphisms and H 2 stands for the second cohomology group of the complex . In this paper, which is mainly expository in nature, we reexamine Harrison’s proof and show how [8] connects with Kummer theory and the theory of galois algebras [16]. We emphasize that most of the ideas on which this paper is based originate in [8].

On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and VA be the k-subalgebra of Endk(A) generated by EndA(A) = A and all k-derivations of A. A study of the homological properties of VA was initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Bjork [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of VA is between r and 2r. Moreover, if k has positive characteristic, then gl.dim VA = 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim VA = r if k has characteristic zero and A = k[x1, …, xr]; in this case VA is the so-called “Weyl algebra on 2r variables”.

Ramification invariants and torsion galois module structure in number fields

Let li be a number field with ring of integers o, and K be a normal extension of k with Galois group r. The integer ring .O of K is then a Gaiois module. i.e.. a module over the group ring 01’. During the last few years a great deal of information has been obtained regarding the structure of such Galois modules. In particular, Friihlich’s conjecture-recently proved by Taylor [ 141 -asserts that if K/k is tame then C is determined up to s:able ;I-module isomorphism by the Artin root numbers of the symplectic characters of I’. with % the ring of rational integers. The theory, as developed by Fr(ihlich 15 I. Taylor [ 141, and others. encompasses three distinct but apparently related topics in algebraic number theory: (1) Galois module structure of rings of algebraic integers. (2) Galois module structure of ideal class groups of number fields and Grothcndieck and class groups of group rings. and (3) the study of arithmetically or analytically defined functions on complex characters of the Galois group such as the Artin root number. the Artin conductor. and the Galois Gauss sum. In this paper we investigate two other Galois modules which intervene ip a natural way in the consideration of the above topics. These modules arise as elementary ramification invariants of the extension K/k. Their distinctive feature is that they arc torsion o-modules. Although they are completeiy determined by local data. their projective resoiutions, together with standard applications of Schanuel’s Lemma [ 1, Proposition 6.3, p. 361. yield information on global Galois module structure. The simplest of these modules is defined by the equality

ON THE AUTOMORPHISM SCHEME OF A PURELY INSEPARABLE FIELD EXTENSION

Publisher Summary This chapter describes the automorphism scheme of a purely inseparable field extension. It discusses the automorphism scheme of a finite field extension and its significance in field theory. The chapter discusses the automorphism scheme of K / k for the special case in which k has characteristic p ≠ 0 and K is purely inseparable and modular over k , isomorphic to a tensor product over k of primitive extensions. It presents a number of elementary and essentially well-known facts about automorphism schemes. It is proved that if K is a tensor product over k of primitive purely inseparable extensions of k of equal degree, then A ( K / k ) is an integral domain and its quotient field is isomorphic to a pure function field over K . In particular, the automorphism scheme of K / k is, in this case, irreducible and reduced. This is not true for an arbitrary purely inseparable modular extension. An extension K / k for which A ( K / k ) possesses nontrivial nilpotent elements is also exhibited.

Abelian Extensions and a Cohomology Theory of Harrison

Let k be a field, K be a separable closure of k, and П be the group of all automorphisms of K leaving k pointwise fixed. П is a compact, totally disconnected group in the topology for which a neighborhood base at the identity is the set of subgroups of П which correspond, via the fundamental theorem of Galois theory, to the finite separable extensions of k.