This paper deals with the Riemann-Roch problem for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. A similar problem for Deligne-Mumford stacks has been solved by B. Toen [\(K\)-Theory 18, 33-76 (1999; Zbl 0946.14004)] using results of A. Vistoli on equivariant higher algebraic \(K\)-theory of regular schemes under the action of a finite group [Duke Math. J. 63, 399-419 (1991; Zbl 0738.55002)]. Let \(G\) be a finite group viewed as a group scheme over a field \(k\), and assume that the order of \(G\) is prime to the characteristic of \(k\). The Grothendieck group of vector bundles on the stack \([\text{Spec} k/G]\) which is denoted by \(K^0_G(\text{Spec} k)\) is isomorphic to the representation ring of \(G\). Let \({\mathcal K}_i\otimes\mathbb{Q}\) be the presheaf: \(U\to K_i(U)\otimes \mathbb{Q}\) associated to the \(K\)-theory groups and let \({\mathcal K}_\mathbb{Q}\) be the corresponding presheaf of the algebraic \(K\)-theory spectrum. The diagram, which should provide “Riemann-Roch” in this context, \[ \begin{matrix} K^0_G(\text{Spec} k) & \to & \mathbb{H}^0_{et}([\text{Spec} k/G], {\mathcal K}_\mathbb{Q})\\ p_*\downarrow & & p_* \downarrow \\ K^0(\text{Spec} k) &\to & \mathbb{H}^0_{et}(\text{Spec} k,{\mathcal K}_\mathbb{Q}) \end{matrix} \] where \(p:[\text{Spec} k/G]\to \text{Spec} k\) is the map of algebraic stacks, does not in general commute.
The main ingredient of this paper is to show that if one replaces the étale topology with the so-called isovariant topology and the presheaf \({\mathcal K}\) with the corresponding equivariant version \({\mathcal K}^G\) then the corresponding diagram commutes. – The Riemann-Roch theorem is established here as a natural transformation between the \(G\)-theory of algebraic stacks and topological \(G\)-theory:
Theorem. Let \(f:{\mathcal S}'\to{\mathcal S}\) denote any proper map between two algebraic stacks being finitely presented and of finite cohomological dimension over a base scheme \(S\) (noetherian and of finite Krull dimension). Then the direct image map \(f_*\) fits into the following homotopy commutative diagram: \[ \begin{matrix} G({\mathcal S}') & \to & G^{\text{top}}({\mathcal S}')\\ f_*\downarrow & & f_*\downarrow\\ G({\mathcal S}) & \to &G^{\text{top}}({\mathcal S})\end{matrix}. \]