Abstract
The purpose of this paper is to prove an equivariant Riemann-Roch theorem for schemes or algebraic spaces with an action of a linear algebraic group
G
. For a
G
-space
X
, this theorem gives an isomorphism between a completion of the equivariant Grothendieck group and a completion of equivariant equivariant Chow groups.
The key to proving this isomorphism is a geometric description of completions of the equivariant Grothendieck group. Besides Riemann-Roch, this result has some purely
K
-theoretic applications. In particular, we prove a conjecture of Köck (in the case of regular schemes) and extend to arbitrary characteristic a result of Segal on representation rings.