In mathematics, the equivariant algebraic K-theory is an algebraic K-theory associated to the category
Coh
G
(
X
)
of equivariant coherent sheaves on an algebraic scheme X with action of a linear algebraic group G, via Quillen's Q-construction; thus, by definition,
K
i
G
(
X
)
=
π
i
(
B
+
Coh
G
(
X
)
)
.
In particular,
K
0
G
(
C
)
is the Grothendieck group of
Coh
G
(
X
)
. The theory was developed by R. W. Thomason in 1980s. Specifically, he proved equivariant analogs of fundamental theorems such as the localization theorem.
Equivalently,
K
i
G
(
X
)
may be defined as the
K
i
of the category of coherent sheaves on the quotient stack
[
X
/
G
]
. (Hence, the equivariant K-theory is a specific case of the K-theory of a stack.)
A version of the Lefschetz fixed point theorem holds in the setting of equivariant (algebraic) K-theory.
