using the derived category of singularities for isolated singularities can be extended to isolated Cohen-Macaulay singularities, giving techniques for computing the Grothendieck group of any singular algebraic curve. This is because reduction gives a generically smooth curve, and all singularities are Cohen-Macaulay. 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 then there is a short exact sequence where is the conormal bundle of in . If we have a singular space embedded into a smooth space we define the virtual conormal bundle as Another useful application of virtual bundles is with the definition of a virtual tangent bundle of an intersection of spaces: Let be projective subvarieties of a smooth projective variety. Then, we can define the virtual tangent bundle of their intersection as Kontsevich uses this construction in one of his papers. Chern characters 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 More generally, if is a direct sum of line bundles, with first Chern classes the Chern character is defined additively 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. Equivariant K-theory The equivariant algebraic K-theory is an algebraic K-theory associated to the category of equivariant coherent sheaves on an algebraic scheme with action of a linear algebraic group , via Quillen's Q-construction; thus, by definition, In particular, is the Grothendieck group of . The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem. See also Bott periodicity KK-theory KR-theory List of cohomology theories Algebraic K-theory Topological K-theory Operator K-theory Grothendieck–Riemann–Roch theorem Notes References External links Grothendieck-Riemann-Roch Max Karoubi's Page K-theory preprint archive