Editing K-theory (section)

=== K<sub>0</sub> of a smooth projective curve ===
For a smooth projective curve <math>C</math> the Grothendieck group is
<math display="block">K_0(C) = \mathbb{Z}\oplus\text{Pic}(C)</math>
for [[Picard group]] of <math>C</math>. This follows from the [[Quillen spectral sequence|Brown-Gersten-Quillen spectral sequence]]<ref name=":1">{{Cite book|last=Srinivas, V.|url=https://www.worldcat.org/oclc/624583210|title=Algebraic K-theory|date=1991|publisher=Birkhäuser|isbn=978-1-4899-6735-0|location=Boston|oclc=624583210}}</ref><sup>pg 72</sup>  of [[algebraic K-theory]]. For a [[regular scheme]] of finite type over a field, there is a convergent spectral sequence
<math display="block">E_1^{p,q} = \coprod_{x \in X^{(p)}}K^{-p-q}(k(x)) \Rightarrow K_{-p-q}(X)</math>
for <math>X^{(p)}</math> the set of codimension <math>p</math> points, meaning the set of subschemes <math>x: Y \to X</math> of codimension <math>p</math>, and <math>k(x)</math> the algebraic function field of the subscheme. This spectral sequence has the property<ref name=":1" /><sup>pg 80</sup>
<math display="block">E_2^{p,-p} \cong \text{CH}^p(X)</math>
for the Chow ring of <math>X</math>, essentially giving the computation of <math>K_0(C)</math>. Note that because <math>C</math> has no codimension <math>2</math> points, the only nontrivial parts of the spectral sequence are <math>E_1^{0,q},E_1^{1,q}</math>, hence
<math display="block">\begin{align}
E_\infty^{1,-1}\cong E_2^{1,-1} &\cong \text{CH}^1(C) \\
E_\infty^{0,0} \cong E_2^{0,0} &\cong \text{CH}^0(C)
\end{align}</math>
The [[coniveau filtration]] can then be used to determine <math>K_0(C)</math> as the desired explicit direct sum since it gives an exact sequence
<math display="block">0 \to F^1(K_0(X)) \to K_0(X) \to K_0(X)/F^1(K_0(X)) \to 0</math>
where the left hand term is isomorphic to <math> \text{CH}^1 (C) \cong \text{Pic}(C)</math> and the right hand term is isomorphic to <math>CH^0(C) \cong \mathbb{Z}</math>. Since <math>\text{Ext}^1_{\text{Ab}}(\mathbb{Z},G) = 0</math>, we have the sequence of abelian groups above splits, giving the isomorphism. Note that if <math>C</math> is a smooth projective curve of genus <math>g</math> over <math>\mathbb{C}</math>, then
<math display="block">K_0(C) \cong \mathbb{Z}\oplus(\mathbb{C}^g/\mathbb{Z}^{2g})</math>
Moreover, the techniques above using the derived category of singularities for isolated singularities can be extended to isolated [[Cohen–Macaulay ring|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.