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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
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1 Answer 1

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.

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Thank you a lot! The deal is that I have made a reduction of my problem to the question in equivarient K-theory but I am not familiar with it and I cant find an article with basic definitions –  user11335 Dec 5 '10 at 20:38
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Sure. One place to go for basic definitions is the Chapter 5 of the book by Chriss and Ginzburg. (Unfortunately there's no index, but that chapter is pretty easy to navigate.) –  Dave Anderson Dec 5 '10 at 20:43
    
Thank you so much! I will now search for it. –  user11335 Dec 5 '10 at 20:46
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The article of Merkurjev in the Handbook of K-theory is also quite useful: it summarizes basic useful properties. –  Baptiste Calmès Dec 6 '10 at 11:28
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