Richard G. Swan

Vector bundles and projective modules

Serre [9, ?50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring. For some time, it has been assumed thlat a similar correspondence exists between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way.

The Whitehead group of a polynomial extension

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K-Theory of Finite Groups and Orders

Frobenius functors.- Finiteness theorems.- Results on K0 and G0.- Maximal orders.- Orders.- K0 of a maximal order.- K1 and G1.- Cancellation theorems.

The Grothendieck ring of a finite group

IF E is any additive category, the Grothendieck group Kc%?) is defined to be the group with one generator [A] for each object A of V and relations [A] = [A’] + [A”] whenever there is an exact sequence 0 + A’ + A -+ A” -+ 0 in %’ [6], [23]. If R is a commutative ring and rr is a finite group, I will denote by G(Rrr) the Grothendieck group of the category of finitely generated: Rn-modules [23]. The main purpose of this paper is to obtain information about the structure of G(Rn) when R is a Dedekind ring.

Topological examples of projective modules

A new and more elementary proof is given for Lonsteds theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of Lensted. In certain cases, methods associated with Huberts 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If m = 2 mod 4, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of rank # m are free. In [29], Lonsted proved a remarkable theorem which shows that vector bundles over a finite CW complex can be represented by finitely generated projective modules over a noetherian ring. This means that by purely topological constructions one can produce examples of noetherian rings whose projective modules have certain specified properties. This method has the advantage that one can impose conditions on the totality of projective modules while the more elementary method of [42] only allows us to construct a finite number of such modules at a time. Lonsteds construction makes use of rather deep properties of analytic functions. In attempting to analyze his proof, I discovered a more elementary proof of the theorem which I will present here. This proof gives rings which are considerably smaller than the ones used by Lonsted. They are, in fact, localizations of algebras of finite type over R. In certain cases, one can even give a very simple and purely algebraic description of the ring. This will be done in §10 using methods associated with Huberts 17th problem. The starting point for this work was a question of A. Geramita. He pointed out that the rings in [42] have nontrivial projective modules of low rank and asked whether, for all n, there are noetherian rings having nontrivial projective modules but such that all such modules of rank less than n are free. Three Received by the editors December 9, 1975. AMS iMOS) subject classifications (1970). Primary 13C10, 55F25, 12D15; Secondary 16A50, 55F50.

Minimal resolutions for finite groups

(1) ~~*+Fz-+F,+F,+Z--+O where G acts trivially on Z and each F, is ZG-free. For computation of the cohomology of G it is convenient to choose a resolution (1) in which the Fi do not have too many generators. I will consider here the following general problem: Suppose we are given a sequence of integers fO, fi, .... What conditions must the fi satisfy in order that there exists a resolution (1) in which Fi is ZG-free on fi generators? The main result will be an almost complete solution of this problem for the case in which G is finite. The conditions on thefi will be a set of inequalities connecting them with cohomological invariants of G. If G is a finite p-group, our theorem implies that there is a resolution (1) in whichfi = dim H’(G, Z,). Therefore G has a resolution which is minimal in an obvious sense. Before proving the main theorem I will give a number of necessary conditions which thefi must satisfy. These conditions do not require the group G to be finite. Since the conditions are closely related to the Morse inequalities, it will be convenient to make the following definition DEFINITION. Let F be a resolution ~~~-+F2--+F1-+F,,--+Z--+0

Noether’s Problem in Galois Theory

This paper is essentially an expanded version of [Sw] and gives a more detailed discussion of some of the topics mentioned there. It is intended to serve as an introduction to recent work related to Noether’s problem. To avoid too much overlap with [Sw] a number of historical remarks, comments, and other topics have been omitted here.

Hochschild cohomology of quasiprojective schemes

Three definitions for the Hochschild cohomology of schemes are considered and shown to coincide for quasiprojective schemes. In the smooth case, the associated Hodge spectral sequences are also shown to be isomorphic.

Vector Bundles over Affine Surfaces

In view of the well-known correspondence between vector bundles and projective modules [50], this implies a similar cancellation theorem for vector bundles on an affine surface over an algebraically closed field. The example of the tangent bundle of the real 2-sphere [57] shows that Theorem 1 fails for k = IR. We will consider some cases where it holds for non-algebraically closed fields in w 3. We can use this theorem to obtain a result relating the structure of vector bundles to more geometric properties of the surface. We let K0(A)=ker[ rk : Ko(A)---, Z], SKo(A) = ker [det: Ko(A ) ~ Pic A] and SA o(X) = ker [Ao(X)--, Alb X], where A o is the group of zero cycles of degree zero modulo rational equivalence [36, 41, 42], and Alb X is the Albanese variety of X.

AN APPLICATION OF GRAPH THEORY TO ALGEBRA

[Al, * * * , Ak ] 54-0. The original proof of the theorem [1] was elementary but very complicated. In attempting to simplify this proof, I found a more transparent proof based on the use of graph theory.3 One advantage of this approach is that complicated algebraic definitions can be replaced by much simpler geometric definitions merely by drawing a picture of the appropriate graph. Before stating the graph theoretic theorem which implies Theorem 1, I will give some elementary definitions and lemmas from graph theory.