be functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that Also, if
is a (global) local complete intersection morphism; i.e., it factors as a closed regular embedding
, then let 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 then amounts to the construction of a unique natural transformation:[1] between the two functors such that for each scheme X in C, the homomorphism
satisfies: for a local complete intersection morphism
Aside from algebraic spaces, no straightforward generalization is possible for stacks.
The complication already appears in the orbifold case (Kawasaki's Riemann–Roch).
The equivariant Riemann–Roch theorem for finite groups is equivalent in many situations to the Riemann–Roch theorem for quotient stacks by finite groups.
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.