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The K-group $K^T(X)$ of $T$(torus)-equivariant coherent sheaves on a variety $X$ is isomorphic to $K^T(X^T)$, that of the fixed point locus via the inclusion homomorphism, when we tensor the quotient field of the representation ring $R(T)$ of $T$.

Is there a similar result for $D_T(Coh X)$ and $D_T(Coh X^T)$, derived categories of $T$-equivariant coherent sheaves ? I do not know even how to formulate `the quotient field of $R(T)$' in the derived category case.

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This seems like exactly the sort of thing that ought to be one of the main results in the paper of Goresky-Kottwitz-MacPherson. Unfortunately, I haven't been able to extract the kind of statement you're asking for; it would be very interesting if someone can give a precise answer. They do give a localization-type theorem for an element of the equivariant derived category in (7.8), but only after taking cohomology. –  Dave Anderson Dec 6 '10 at 0:59
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It doesn't seem like that to me at all. Is there anything about coherent sheaves in GKM? –  Ben Webster Dec 6 '10 at 7:53
    
@Ben, you're absolutely right, somehow I missed the salient word "coherent" in the question. But even for constructible sheaves, is there a localization statement at the level of derived categories? –  Dave Anderson Dec 6 '10 at 16:02

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