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Riemann-Roch for equivariant Chow groups. (English) Zbl 0997.14002

In this paper, the authors prove an equivariant version of the Riemann-Roch theorem for an algebraic space \(X\) with an action of a linear algebraic group \(G\). Consider the Grothendieck group of \(G\)-equivariant coherent sheaves on \(X\) tensored by the field of rational numbers. The theorem says that there is an isomorphism between the completion of this group along the augmentation ideal and the product of all the equivariant rational Chow groups of \(X\). In order to prove the theorem, a geometric description of completions of the equivariant Grothendieck group is given.
Two applications involving \(K\)-theory are given. First, a conjecture of Köck is proven for the case where \(X\) is a regular scheme over a field. Secondly, a result of Segal on representation rings is extended to arbitrary characteristic. Both these results are a consequence of a theorem that states that the completions of the equivariant \(K\)-theory with respect to two specific ideals are isomorphic.

MSC:

14C40 Riemann-Roch theorems
19L47 Equivariant \(K\)-theory
14L30 Group actions on varieties or schemes (quotients)
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References:

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