# 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: ${\displaystyle G_{*},A_{*}}$ are functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that

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

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

${\displaystyle T_{f}=[T_{P/Y}|_{X}]-[N_{X/P}]}$

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:[1]

${\displaystyle \tau :G_{*}\to A_{*}}$

between the functors such that

• for ${\displaystyle \tau _{X}\circ f^{*}=\operatorname {td} (T_{f})\cdot f^{*}\circ \tau _{Y}}$, where we wrote ${\displaystyle \tau _{X}:G(X)\to A(X)}$ and ${\displaystyle \operatorname {td} (T_{f})}$ 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,

## References

1. ^ Fulton, Theorem 18.3.

## References

• Edidin, Dan (2012-05-21). "Riemann-Roch for Deligne-Mumford stacks". arXiv:1205.4742 [math.AG].
• William Fulton (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 2 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-62046-4, MR 1644323
• Toen, B. (1998-03-17). "Riemann-Roch Theorems for Deligne-Mumford Stacks". arXiv:math/9803076.
• Bertrand, Toen (1999-08-18). "K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems". arXiv:math/9908097.
• Lowrey, Parker; Schürg, Timo (2012-08-30). "Grothendieck-Riemann-Roch for derived schemes". arXiv:1208.6325 [math.AG].