Editing K-theory (section)

==Applications==
===Virtual bundles===
One useful application of the Grothendieck-group is to define virtual vector bundles. For example, if we have an embedding of smooth spaces <math>Y \hookrightarrow X</math> then there is a short exact sequence

:<math> 0 \to \Omega_Y \to \Omega_X|_Y \to C_{Y/X} \to 0</math>

where <math>C_{Y/X}</math> is the conormal bundle of <math>Y</math> in <math>X</math>. If we have a singular space <math>Y</math> embedded into a smooth space <math>X</math> we define the virtual conormal bundle as

:<math>[\Omega_X|_Y] - [\Omega_Y]</math>

Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let <math>Y_1,Y_2\subset X</math> be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection <math>Z = Y_1\cap Y_2</math> as

:<math> [T_Z]^{vir} = [T_{Y_1}]|_Z + [T_{Y_2}]|_Z - [T_{X}]|_Z.</math>

Kontsevich uses this construction in one of his papers.<ref>{{citation |last1=Kontsevich |first1=Maxim |author-link=Maxim Kontsevich| contribution=Enumeration of rational curves via torus actions |arxiv=hep-th/9405035|year=1995|title=The moduli space of curves (Texel Island, 1994)|pages=335–368|series=Progress in Mathematics|volume= 129|publisher= Birkhäuser Boston|location= Boston, MA|mr=1363062}}</ref>

===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]].