Editing K-theory (section)

=== Grothendieck group of coherent sheaves in algebraic geometry ===
In algebraic geometry, the same construction can be applied to algebraic vector bundles over a smooth scheme. But, there is an alternative construction for any Noetherian scheme <math>X</math>. If we look at the isomorphism classes of [[Coherent sheaf|coherent sheaves]] <math>\operatorname{Coh}(X)</math> we can mod out by the relation <math>[\mathcal{E}] = [\mathcal{E}'] + [\mathcal{E}'']</math> if there is a [[Exact sequence|short exact sequence]]

:<math>0 \to \mathcal{E}' \to \mathcal{E} \to \mathcal{E}'' \to 0.</math>

This gives the Grothendieck-group <math>K_0(X)</math> which is isomorphic to <math>K^0(X)</math> if <math>X</math> is smooth. The group <math>K_0(X)</math> is special because there is also a ring structure: we define it as

:<math>[\mathcal{E}]\cdot[\mathcal{E}'] = \sum(-1)^k \left [\operatorname{Tor}_k^{\mathcal{O}_X}(\mathcal{E}, \mathcal{E}') \right ].</math>

Using the [[Grothendieck–Riemann–Roch theorem]], we have that

:<math>\operatorname{ch} : K_0(X)\otimes \Q \to A(X)\otimes \Q</math>

is an isomorphism of rings. Hence we can use <math>K_0(X)</math> for [[intersection theory]].<ref>{{Cite web|last=Grothendieck|title=SGA 6 - Formalisme des intersections sur les schema algebriques propres|url=http://library.msri.org/books/sga/sga/6/6t_519.html|access-date=2020-10-20|archive-date=2023-06-29|archive-url=https://web.archive.org/web/20230629053130/http://library.msri.org/books/sga/sga/6/6t_519.html|url-status=dead}}</ref>