A formal difference (with respect to direct sum/Whitney sum) of vector bundles. Appears as representative for classes in topological K-theory. A virtual vector bundle has a virtual rank , which is the integer .
Let X be a space that is locally compact and regular. Then any virtual vector bundle of virtual rank 0 has some tensor power that is 0 in K-theory.
The following argument comes from (Karoubi 1978, Theorem II.5.9), but generalised to take into account non-compact spaces which are introduced in the following pages.
For closed subsets and , then the ring structure on K-theory means
where . We can likewise consider the same for a sequence .
If is in (i.e. reduced K-theory ) with in some compact set (recall that vector bundle K-theory elements are differences of vector bundles that are isomorphic outside a compact set. Without loss of generality such an element can be represented as where is a trivial bundle with fibre ), cover by finitely many closed neighbourhoods over which trivialises. Thus is in for each .
Thus is in the intersection of all the as well as for the complement of . Thus is in .
Hence for every element in , some power of it is zero, hence is a nil ideal.
Last revised on August 18, 2016 at 08:40:58. See the history of this page for a list of all contributions to it.