Editing K-theory (section)

==Definitions==
{{About| K<sub>0</sub>, the most basic K-theory group (see also [[Grothendieck group]])|definitions of higher K-groups K<sub>i</sub>|Algebraic K-theory|and|Topological K-theory|section=yes}}
There are a number of basic definitions of K-theory: two coming from topology and two from algebraic geometry.

=== Grothendieck group for compact Hausdorff spaces ===
Given a compact [[Hausdorff space]] <math>X</math> consider the set of isomorphism classes of finite-dimensional vector bundles over <math>X</math>, denoted <math>\text{Vect}(X)</math> and let the isomorphism class of a vector bundle <math>\pi:E \to X</math> be denoted <math>[E]</math>. Since isomorphism classes of vector bundles behave well with respect to [[direct sum]]s, we can write these operations on isomorphism classes by

:<math>[E]\oplus[E'] =[E\oplus E'] </math>

It should be clear that <math>(\text{Vect}(X),\oplus)</math> is an abelian monoid where the unit is given by the trivial vector bundle <math>\R^0\times X \to X</math>. We can then apply the Grothendieck completion to get an abelian group from this abelian monoid. This is called the K-theory of <math>X</math> and is denoted <math>K^0(X)</math>.

We can use the [[Serre–Swan theorem]] and some algebra to get an alternative description of vector bundles over <math>X</math> as [[projective module]]s over the ring <math>C^0(X;\Complex)</math> of continuous complex-valued functions. Then, these can be identified with [[Idempotence|idempotent]] matrices in some ring of matrices <math>M_{n\times n}(C^0(X;\Complex))</math>. We can define equivalence classes of idempotent matrices and form an abelian monoid <math>\textbf{Idem}(X)</math>. Its Grothendieck completion is also called <math>K^0(X)</math>. One of the main techniques for computing the Grothendieck group for topological spaces comes from the [[Atiyah–Hirzebruch spectral sequence]], which makes it very accessible. The only required computations for understanding the spectral sequences are computing the group <math>K^0</math> for the spheres <math>S^n</math>.<ref>{{Cite book|last=Park, Efton.|url=https://www.worldcat.org/oclc/227161674|title=Complex topological K-theory|date=2008|publisher=Cambridge University Press|isbn=978-0-511-38869-9|location=Cambridge|oclc=227161674}}</ref><sup>pg 51-110</sup>

=== Grothendieck group of vector bundles in algebraic geometry ===
There is an analogous construction by considering vector bundles in [[algebraic geometry]]. For a [[Noetherian scheme]] <math>X</math> there is a set <math>\text{Vect}(X)</math> of all isomorphism classes of [[algebraic vector bundle]]s on <math>X</math>. Then, as before, the direct sum <math>\oplus</math> of isomorphisms classes of vector bundles is well-defined, giving an abelian monoid <math>(\text{Vect}(X),\oplus)</math>. Then, the Grothendieck group <math>K^0(X)</math> is defined by the application of the Grothendieck construction on this abelian monoid.

=== 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>