of a projective bundle Another important formula for the Grothendieck group is the projective bundle formula: given a rank r vector bundle over a Noetherian scheme , the Grothendieck group of the projective bundle is a free -module of rank r with basis . This formula allows one to compute the Grothendieck group of . This make it possible to compute the or Hirzebruch surfaces. In addition, this can be used to compute the Grothendieck group by observing it is a projective bundle over the field . K0 of singular spaces and spaces with isolated quotient singularities One recent technique for computing the Grothendieck group of spaces with minor singularities comes from evaluating the difference between and , which comes from the fact every vector bundle can be equivalently described as a coherent sheaf. This is done using the Grothendieck group of the Singularity category from derived noncommutative algebraic geometry. It gives a long exact sequence starting with where the higher terms come from higher K-theory. Note that vector bundles on a singular are given by vector bundles on the smooth locus . This makes it possible to compute the Grothendieck group on weighted projective spaces since they typically have isolated quotient singularities. In particular, if these singularities have isotropy groups then the map is injective and the cokernel is annihilated by for .pg 3 K0 of a smooth projective curve For a smooth projective curve the Grothendieck group is for Picard group of . This follows from the Brown-Gersten-Quillen spectral sequencepg 72 of algebraic K-theory. For a regular scheme of finite type over a field, there is a convergent spectral sequence for the set of codimension points, meaning the set of subschemes of codimension , and the algebraic function field of the subscheme. This spectral sequence has the propertypg 80 for the Chow ring of , essentially giving the computation of . Note that because has no codimension points, the only nontrivial parts of the spectral sequence are , hence The coniveau filtration can then be used to determine as the desired explicit direct sum since it gives an exact sequence where the left hand term is isomorphic to and the right hand term is isomorphic to . Since , we have the sequence of abelian groups above splits, giving the isomorphism. Note that if is a smooth projective curve of genus over , then Moreover, the techniques above