What is the construction of pushforward in algebraic equivarient Ktheory ?
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It's the same as in nonequivariant $K$theory. For a $G$equivariant proper morphism $f:X \to Y$ and an equivariant coherent sheaf $F$ on $X$, define $$f^G_*[F] = \sum (1)^i [R^i f_*F],$$ which makes sense because each higher direct image is equivariant (and coherent, because $f$ is proper). An interesting situation is when $Y$ is a point, in which case this is the "equivariant Euler characteristic," an alternating sum of (virtual) $G$representations. 

