Editing K-theory (section)

=== K<sub>0</sub> 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 <math>K^0(X)</math> and <math>K_0(X)</math>, 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]] <math>D_{sg}(X)</math><ref>{{Cite web|title=ag.algebraic geometry - Is the algebraic Grothendieck group of a weighted projective space finitely generated ?|url=https://mathoverflow.net/questions/133383/is-the-algebraic-grothendieck-group-of-a-weighted-projective-space-finitely-gene|access-date=2020-10-20|website=MathOverflow}}</ref><ref name=":0">{{cite journal|last1=Pavic|first1=Nebojsa|last2=Shinder|first2=Evgeny|title=K-theory and the singularity category of quotient singularities|journal=Annals of K-Theory|year=2021|volume=6|issue=3|pages=381–424|doi=10.2140/akt.2021.6.381|arxiv=1809.10919|s2cid=85502709}}</ref> from [[derived noncommutative algebraic geometry]]. It gives a long exact sequence starting with
<math display="block">\cdots \to K^0(X) \to K_0(X) \to K_{sg}(X) \to 0</math>
where the higher terms come from [[Algebraic K-theory|higher K-theory]]. Note that vector bundles on a singular <math>X</math> are given by vector bundles <math>E \to X_{sm}</math> on the smooth locus <math>X_{sm} \hookrightarrow X</math>. 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 <math>G_i</math> then the map
<math display="block">K^0(X) \to K_0(X)</math>
is injective and the cokernel is annihilated by <math>\text{lcm}(|G_1|,\ldots, |G_k|)^{n-1}</math> for <math>n = \dim X</math>.<ref name=":0" /><sup>pg 3</sup>