R. Keith Dennis

Units of group rings

Abstract Let G be an abelian group and let Z G be its integral group ring. In special cases (e.g., G finite [4]) it has been noted that G is a direct summand of ( Z G)∗, the group of units of Z G. However, no explicit construction for the splitting maps is given in the literature. In Section 1 it will be shown that for any abelian group G, there is a canonical splitting of the inclusion G→ Z G∗. Upon attempting to generalize this result to other coefficient rings, we are led to the concept of a semimodule. We show in Section 2 that G→(AG)∗ splits if G admits a semimodule structure over A. The final section contains a number of partial results and examples showing the difficulty of deciding whether or not splittings exist in the general case. This paper originated from the construction of maps in algebraic K-theory using the Hochschild homology of a bimodule [6, Chap.X]. Let G→GLn Z G be given by gdiag (g, 1,…, 1). It will be shown elsewhere that for an arbitrary group G, there is a canonical splitting of the induced map Hi(G; Z )→Hi(GLn( Z G); Z ) (ordinary homology of groups with trivial action on Z for all i ⩾ 0, n ⩾ 1. As the case i = n = 1 is of independent interest, we present it here rather than obscure it in a paper dealing with algebraic K-theory. For a given integer i, the splitting is constructed via properties of Hochschild homology groups Hi(R, R) for certain rings R. For i = 1 and R a commutative ring, H1(R, R) is canonically isomorphic to Ω 1 R Z , the module of Kahler differentials. In Section 1 we use Ω 1 R Z and its properties to construct splittings as it will be more familiar than Hochschild homology groups to most readers.

Injective stability for

It has been conjectured for some time [St2], [D], [S-D], [D-S2] that for a semilocal ring R, the homomorphisms On:K2(n, R)-+K2(n + l, R), known to be surjective for all n^.2 [St2], [S-D], are in fact isomorphisms for n^.3. Various special cases have been proved, most notably the difficult theorem of Matsumoto for fields [Ma, Corollaire 5.11] and the case of discrete valuation rings [D-Sl]. Matsumoto also shows that 62 is not an isomorphism in general. In this note we announce the proof of the following theorem, details of which will be published elsewhere. Unexplained notation and terminology is that of [Mi].

K_2

[G]) which can be used both for specific computations and to prove general theo- rems for certain classes of groups [D, St2]. In collaboration with Oliver we have greatly extended the power of our computational methods; the results of all computations as well as the ad- ditional methods leading to them will appear in the sequel [A-D-O-S] to this paper in a future issue of Inventiones. There we derive a very efficient algo- rithm for computing

of local rings

This chapter is concerned with the classification of finite dimensional central division algebras over a given field k. In the case k = R, the Frobenius Theorem shows that R and H are the only finite dimensional central division algebras over R. This kind of classification is optimal in the sense that we have an explicit, easy-to-understand list of all finite dimensional central division algebras over R. Classifying finite dimensional central division algebras over other fields has proven much more difficult, and in fact this problem has been a focal point for research in number theory and quadratic forms. Although such an explicit list as in the case of central division algebras over R cannot always be given, there is much that can be said.

SK1 of finite abelian groups, II

We saw in Theorem 1.15 that simple artinian rings are precisely those artinian rings which have a faithful simple module. It is useful to drop the finiteness condition and to study those rings which have a faithful simple module but are not necessarily artinian. Such a ring is called a primitive ring. Primitive rings, a generalization of simple rings, play a role analogous to that of simple rings in that they may be viewed as the basic building blocks of other rings, though in an extended, infinite dimensional context. This perhaps justifies the name primitive. The theory of primitive rings can be developed along lines parallel to that of simple rings. The two theories intertwine, and in fact some authors choose to study simple rings from the point of view of primitive rings. This chapter explores such an approach.

The Brauer Group

This chapter is concerned with looking at part of a structure theory for rings. The idea of any “structure theory” of an object (in this case a ring) is to express that object in terms of simpler, better understood pieces. For example, the Wedderburn Structure Theorem says that any semisimple ring (we’ll define this later) is isomorphic to a finite product of matrix rings over division rings, each of which is simple. The theory for semisimple modules is in many ways analogous to the theory of vector spaces over a field, where we can break up vector spaces as sums of certain subspaces.

Primitive Rings and the Density Theorem

There is an invariant of rings called the global dimension. Semisimple rings are precisely those rings with global dimension zero. Thus the material in Chapters 1 and 2 can be considered the zero’th step in the theory of global dimension. Kaplansky, based upon an observation of Schanuel, was the first to set down the dimension theory of rings in an elementary way, without using the powerful machinery of homological algebra. This section is based on his Queen Mary College notes.

Semisimple Modules & Rings and the Wedderburn Structure Theorem

The Brauer group Br(R) of a commutative ring was introduced by Auslander and Goldman in their 1960 paper The Brauer Group of a Commutative Ring, building on earlier work of Azumaya. This group coincides with the “classical” Brauer group (cf. Chapter 4) in the case when R is a field. One of the points of extending the theory to rings is that one can relate Brauer groups of fields to Brauer groups of related rings in exact sequences; one then hopes that this will help compute the classical Brauer group. The Brauer group of a commutative ring is also part of a Galois theory of commutative rings. For more on these matters, the reader may consult Galois Theory and Cohomology of Commutative Rings by Chase, Harrison and Rosenberg, The Brauer Group of Commutative Rings by Orzech and Small, Separable Algebras Over Commutative Rings by DeMeyer and Ingraham, or the paper of Auslander and Goldman quoted above.

The Global Dimension of a Ring

In this chapter we provide an application of the structure theory of rings developed in Chapters One and Two to the theory of finite groups. Representation theory of finite groups is a vast subject; in this chapter we’ll make a thin beeline right to a famous theorem of Burnside. For a more thorough introduction to the representation theory of finite groups, the reader may consult Serre, Linear Representations of Finite Groups, as well as Fulton and Harris, Representation Theory : A First Course.

The Brauer Group of a Commutative Ring

In Chapter One we developed a structure theory for semisimple rings, as summarized in Theorem 1.18. This theory used, for the most part, properties of modules over a semisimple ring in order to characterize such a ring. In this chapter, we give a more intrinsic characterization of semisimple rings.