# The construction of push-forward in algebraic equivarient K-theory

What is the construction of push-forward in algebraic equivarient K-theory ?

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It is what I expect but I have problems in verifying that it is a correct construction. – user11335 Dec 5 '10 at 20:22
Do you mean why is it well-defined? If so, it's also the same reason as in ordinary K-theory, just use the long exact sequence. (Hopefully that helps, but if not, maybe you could be more specific about the point of confusion?) – Dave Anderson Dec 5 '10 at 20:37

It's the same as in non-equivariant $K$-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.