Draft:Riemann–Roch-type theorem

(The article discusses various generalizations of the Riemann–Roch theorem.)

Formulation due to Baum, Fulton and MacPherson

We need a few notations: $G_{*},A_{*}$ are functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that

• $G_{*}(X)$ is the Grothendieck group of coherent sheaves on X,
• $A_{*}(X)$ is the rational Chow group of X,
• for each proper morphism f, $G_{*}(f),A_{*}(f)$ are the direct image and push-forward along f, respectively.

Also, if $f:X\to Y$ is a local complete intersection morphism; i.e., it factors as a closed regular embedding $X\hookrightarrow P$ into a smooth scheme P followed by a smooth morphism $P\to Y$ , then let

$T_{f}=[T_{P/Y}|_{X}]-[N_{X/P}]$ be the class in the Grothendieck group of vector bundles on X; it is independent of the factorization and is called the virtual tangent bundle of f.

Then the Riemann–Roch theorem amounts to the construction of a unique natural transformation:

$\tau :G_{*}\to A_{*}$ between the functors such that

• for $\tau _{X}\circ f^{*}=\operatorname {td} (T_{f})\cdot f^{*}\circ \tau _{Y}$ , where we wrote $\tau _{X}:G(X)\to A(X)$ and $\operatorname {td} (T_{f})$ is the Todd class of the virtual tangent sheaf.

The Riemann–Roch theorem for Deligne–Mumford stacks

Aside from algebraic spaces, no straightforward generalization is possible for stacks. The complication already appears in the orbifold case (Kawasaki's Riemann–Roch).

One of the significant applications of the theorem is that it allows one to define a virtual fundamental class in terms of the K-theoretic virtual fundamental class. More precisely,