Editing Vector bundle (section)

==K-theory==
The K-theory group, {{math|''K''(''X'')}}, of a compact [[Hausdorff space|Hausdorff]] topological space is defined as the [[abelian group]] generated by [[isomorphism class]]es {{math|[''E'']}} of [[complex vector bundle]]s modulo the [[Relation (mathematics)|relation]] that, whenever we have an [[exact sequence]]
<math display = block> 0 \to A \to B \to C \to 0,</math>
then 
<math display = block>[B] = [A] + [C] </math> in [[topological K-theory]]. [[KO-theory]] is a version of this construction which considers real vector bundles. K-theory with [[compact support]]s can also be defined, as well as higher K-theory groups.

The famous [[Bott periodicity|periodicity theorem]] of [[Raoul Bott]] asserts that the K-theory of any space {{math|''X''}} is isomorphic to that of the {{math|''S''<sup>2</sup>''X''}}, the double suspension of {{math|''X''}}.

In [[algebraic geometry]], one considers the K-theory groups consisting of [[coherent sheaf|coherent sheaves]] on a [[scheme (mathematics)|scheme]] {{math|''X''}}, as well as the K-theory groups of vector bundles on the scheme with the above [[equivalence relation]]. The two constructs are naturally isomorphic provided that the underlying scheme is [[smooth morphism|smooth]].