Editing K-theory (section)

===Chern characters===
{{main|Chern character}}
[[Chern classes]] can be used to construct a homomorphism of rings from the [[topological K-theory]] of a space to (the completion of) its rational cohomology. For a line bundle ''L'', the Chern character ch is defined by

:<math>\operatorname{ch}(L) = \exp(c_{1}(L)) := \sum_{m=0}^\infty \frac{c_1(L)^m}{m!}.</math>

More generally, if <math>V = L_1 \oplus \dots \oplus L_n</math> is a direct sum of line bundles, with first Chern classes <math>x_i = c_1(L_i),</math> the Chern character is defined additively

:<math> \operatorname{ch}(V)  = e^{x_1} + \dots + e^{x_n} :=\sum_{m=0}^\infty \frac{1}{m!}(x_1^m + \dots + x_n^m). </math>

The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. The Chern character is used in the [[Hirzebruch–Riemann–Roch theorem]].