Algebraic Vector Bundles

Abstract

In the first section, we define algebraic R-vector bundles over an affine real algebraic variety X; the definition requires that an algebraic vector bundle be algebraically isomorphic to a subbundle of a trivial bundle. There are several justifications for this requirement. One such justification is that it results in an equivalence of the category of algebraic R-bundles with the category of projective modules of finite type over the ring ℛ(X) of regular functions on X. Section 2 reviews some basic facts concerning the divisor class group of a ring, with some applications to the question of the factoriality of R(X). In Sections 3–6, we are mainly concerned with vector bundles over compact nonsingular algebraic subsets of &#xℝ;ⁿ. In Section 3, we use the Stone-Weierstrass theorem to compare algebraic and topological vector bundles. In Section 4, we study algebraic line bundles and we explain the relation between the algebraic approximation of C ^∞ hypersurfaces of X and the group (math)(X, ℤ/2). Section 5 contains a characterization of those topological vector bundles over X which are isomorphic to algebraic vector bundles, in the case that X is a compact nonsingular algebraic curve or surface. This allows one to compare the algebraic K-theory of ℛ(X) with the topological K-theory of X. The group (math)(X, ℤ/2) plays an important role in this context. In Section 6 we study algebraic ℂ-vector bundles. The results of this section will be particularly useful in Chap. 13. Finally, Section 7 is devoted to Nash and semi-algebraic vector bundles. The main tool here is Efroymson’s approximation theorem. We obtain a purely topological characterization of the affine Nash manifolds whose ring of Nash functions is factorial.

In this chapter we assume knowledge of the basic facts concerning topological or C ^∞ vector bundles (cf. [236] or [174]).