Editing K-theory (section)

==Equivariant K-theory==
The [[equivariant algebraic K-theory]] is an [[algebraic K-theory]] associated to the category <math>\operatorname{Coh}^G(X)</math> of [[equivariant sheaf|equivariant coherent sheaves]] on an algebraic scheme <math>X</math> with [[linear algebraic group action|action of a linear algebraic group]] <math>G</math>,  via Quillen's [[Q-construction]]; thus, by definition,

:<math>K_i^G(X) = \pi_i(B^+ \operatorname{Coh}^G(X)).</math>

In particular, <math>K_0^G(C)</math> is the [[Grothendieck group]] of <math>\operatorname{Coh}^G(X)</math>. The theory was developed by R. W. Thomason in 1980s.<ref>Charles A. Weibel, [https://www.ams.org/notices/199608/comm-thomason.pdf Robert W. Thomason (1952–1995)].</ref> Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.