2 Added EDIT comment at beginning

Ben gave much of the answer, but I'll try to make it precise. Toen says there is no Riemann-Roch for the naive (rational) chow ring (Remark 4.3 in Theoremes de R-R)R-R) (EDIT: unless you also take the naive K-theory).

He says the problem is the chow rings missing the stack structure (p.1), like Ben says. Like you point out, the morphism to a point won't be representable. If a stack has a representable morphism to a scheme, it must be an algebraic space. (Just pull-back by the identity. Representable means this will be an algebraic space.) There's also a problem with the G-theory. The problems are:

• The morphism from a DM-stack to its coarse moduli space $p : F \to M$ induces an isomorphism of rational Chow rings,

$p_{*} : CH(F) \otimes \mathbf{Q} \to CH(M) \otimes \mathbf{Q}$

That's Thm 0.5 from Gillet's intersection theory paper.

• The same morphism induces a weak equivalence (now we're working with simplicial objects)

$p_{*} : G(M) \otimes \mathbf{Q} \to H(F_{et}, G \otimes \mathbf{Q})$

of the cohomology of the G-theory sheaf with the G-theory of the coarse moduli space. (Corollary 3.8 of Toen's R-R paper).

Note that G-theory is formed from the K-groups of the category of coherent sheaves, K-theory from the category of vector bundles. The natural morphism $K \to G$ gives a "Poincare" duality. It is an isomorphism in the case of algebraic spaces, but not in general for stacks (Prop 2.2 of Toen's thesis).

The workaround considers the ramification stack (aka. classifying stack of cyclic subgroups) of F, denoted $I_F$, and has been known in the case of complex orbifolds (V-varieties) since Kawasaki wrote on it in 1979. Toen's proof seems to center on proving

$G_{*}(F) \otimes \mathbf{Q}(\mu_\infty) \cong H^{-*}(I_F, G \otimes \mathbf{Q}(\mu_\infty))$.

I read the left side as the K-theory and the right side as the Chow ring.

He eventually reduces to the known case $F = [X / H]$ of equivariant K-theory with a smooth projective variety $X$ quotiented by a finite group $H$:

$\mathbf{K}_{*}(X, H) \otimes \mathbf{C} \cong \bigoplus_{h \in c(H)} \mathbf{K}_{*}(X^h)^{Z(h)}$,

where the sum is over the set $c(H)$ of conjugacy classes of $H$, $X^h$ is the fixed point subscheme, and $Z(h)$ is the centralizer of $h$ in $H$. (That's Vistoli 1991, maybe also Angeniol, Lejeune-Jalabert 1985)

1

Ben gave much of the answer, but I'll try to make it precise. Toen says there is no Riemann-Roch for the naive (rational) chow ring (Remark 4.3 in Theoremes de R-R).

He says the problem is the chow rings missing the stack structure (p.1), like Ben says. Like you point out, the morphism to a point won't be representable. If a stack has a representable morphism to a scheme, it must be an algebraic space. (Just pull-back by the identity. Representable means this will be an algebraic space.) There's also a problem with the G-theory. The problems are:

• The morphism from a DM-stack to its coarse moduli space $p : F \to M$ induces an isomorphism of rational Chow rings,

$p_{*} : CH(F) \otimes \mathbf{Q} \to CH(M) \otimes \mathbf{Q}$

That's Thm 0.5 from Gillet's intersection theory paper.

• The same morphism induces a weak equivalence (now we're working with simplicial objects)

$p_{*} : G(M) \otimes \mathbf{Q} \to H(F_{et}, G \otimes \mathbf{Q})$

of the cohomology of the G-theory sheaf with the G-theory of the coarse moduli space. (Corollary 3.8 of Toen's R-R paper).

Note that G-theory is formed from the K-groups of the category of coherent sheaves, K-theory from the category of vector bundles. The natural morphism $K \to G$ gives a "Poincare" duality. It is an isomorphism in the case of algebraic spaces, but not in general for stacks (Prop 2.2 of Toen's thesis).

The workaround considers the ramification stack (aka. classifying stack of cyclic subgroups) of F, denoted $I_F$, and has been known in the case of complex orbifolds (V-varieties) since Kawasaki wrote on it in 1979. Toen's proof seems to center on proving

$G_{*}(F) \otimes \mathbf{Q}(\mu_\infty) \cong H^{-*}(I_F, G \otimes \mathbf{Q}(\mu_\infty))$.

I read the left side as the K-theory and the right side as the Chow ring.

He eventually reduces to the known case $F = [X / H]$ of equivariant K-theory with a smooth projective variety $X$ quotiented by a finite group $H$:

$\mathbf{K}_{*}(X, H) \otimes \mathbf{C} \cong \bigoplus_{h \in c(H)} \mathbf{K}_{*}(X^h)^{Z(h)}$,

where the sum is over the set $c(H)$ of conjugacy classes of $H$, $X^h$ is the fixed point subscheme, and $Z(h)$ is the centralizer of $h$ in $H$. (That's Vistoli 1991, maybe also Angeniol, Lejeune-Jalabert 1985)